Asymptotic Theory for Multivariate Nonparametric Quantile Regression with Stationary Ergodic Functional Covariates and Missing-at-Random Responses

Abstract

Quantiles are among the most fundamental constructs in probability theory and statistics, intrinsically linked to order structures, stochastic dominance, and the principles of robust statistical inference. Although the univariate theory of quantiles is by now classical and well developed, their generalization to multivariate settings remains mathematically subtle and methodologically demanding. In particular, extending the notion of “location within a distribution” beyond one dimension raises delicate questions of geometry, ordering, and equivariance. Within this landscape, the spatial—or geometric—formulation of multivariate quantiles has emerged as a rigorous and conceptually unifying framework capable of reconciling these issues. In this work we advance this paradigm by introducing a kernel-based estimation procedure for nonparametric conditional geometric quantiles of a multivariate response $Y\in {\mathbb{R}}^q$ ($q\ge 2$) given a functional covariate X that takes values in an infinite-dimensional space. The data are assumed to form a strictly stationary and ergodic process, while the responses may be subject to a missing-at-random mechanism, a feature of substantial practical relevance. Our analysis establishes strong consistency of the proposed estimator, characterizes its optimal convergence rate, and derives its asymptotic distribution. These limit theorems, in turn, provide the theoretical foundation for constructing asymptotically valid confidence regions and for performing inference in multivariate quantile regression with functional covariates. The theoretical developments rest on natural complexity conditions for the involved functional classes together with mild smoothness and regularity assumptions. This balance between generality and mathematical precision ensures that the resulting methodology is not only robust in a rigorous probabilistic sense but also widely applicable to contemporary problems in high-dimensional and functional data analysis. The proposed methodology is numerically investigated through simulations and is implemented in a real data application.

1. Introduction

Over the past two decades, there has been sustained and rapidly expanding interest in regression frameworks where the response variable is scalar while the covariates are infinite-dimensional objects, typically represented by smooth random functions whose trajectories may vary arbitrarily across observational units. Such observations, commonly referred to as functional data, arise naturally in a wide range of scientific disciplines, including climatology, medicine, economics, and linguistics, where explanatory information is more appropriately modeled as curves or functions rather than finite-dimensional vectors. Comprehensive introductions to the theoretical foundations and methodological tools of functional data analysis (FDA), along with numerous real-world case studies, can be found in [1,2]. It is important to recall that the probabilistic treatment of random elements taking values in normed linear spaces—most notably Banach and Hilbert spaces—substantially predates the modern rise of functional data analysis. As discussed in [3], early developments in probability theory for infinite-dimensional spaces laid much of the theoretical groundwork that now underpins contemporary functional methodologies. Within this broader probabilistic framework, Ref. [4] investigated the intrinsically ill-posed problems of density and mode estimation in normed vector spaces, emphasizing the pronounced impact of the curse of dimensionality when covariates are functional. From the perspective of regression analysis, Ref. [2] represents a seminal contribution to nonparametric modeling with functional predictors, providing a unified treatment of kernel-based methods for estimating conditional quantities such as the regression operator, conditional distribution, and related functionals. Additional foundational works, including [5,6,7], have further shaped the theoretical framework of FDA, offering both rigorous probabilistic foundations and illustrative applications that demonstrate the breadth of the field. More recent research has focused on refining asymptotic theory for conditional functionals in functional settings. In particular, Ref. [8] established rates of uniform consistency for a broad class of functionals of the conditional distribution, uniformly over subsets of the functional covariate space; see also [9,10,11,12]. Building upon these results, Ref. [13] derived uniform-in-bandwidth (UIB) consistency rates for a wide family of nonparametric estimators, including the regression function, conditional distribution function, conditional density, and conditional hazard function. Furthermore, Ref. [14] studied local linear estimation of the regression function in the presence of functional regressors, proving strong convergence uniformly with respect to bandwidth parameters. In a related direction, Ref. [15] investigated k-nearest neighbors estimation for nonparametric regression with strongly mixing functional time series data, establishing uniform nearly complete convergence rates under moderate regularity conditions. For further contemporary developments—covering robust procedures, dependent functional data, and increasingly refined uniform convergence results—the reader may consult [16,17,18,19,20,21,22,23,24,25,26], which collectively illustrate both the methodological maturity and the continuing dynamism of research at the intersection of nonparametric statistics and infinite-dimensional data analysis.

The seminal contribution of [27] triggered a substantial and enduring expansion of interest in quantile regression methodology, owing to its distinctive theoretical and practical advantages. In contrast to classical mean regression, which is concerned with modeling the conditional expectation of a response variable, quantile regression offers a markedly richer inferential framework by characterizing conditional quantiles of the response distribution. This feature provides enhanced flexibility in capturing heterogeneous effects across different parts of the distribution and yields a more robust modeling strategy in the presence of non-Gaussian error structures and outliers. Beyond robustness considerations, quantile regression enables a more comprehensive exploration of the conditional distribution of the response variable by permitting inference on its lower and upper tails, regions that are typically inaccessible through mean-based approaches. As a consequence, it delivers a more nuanced and informative description of the relationship between covariates and response, particularly in situations where distributional asymmetry or tail behavior is of primary interest. A detailed overview of the methodological developments and applied contributions in quantile regression is provided in [28]. The problem of estimating conditional quantiles of a scalar response variable given scalar or multivariate covariates has attracted sustained attention in the statistical literature. Foundational and influential treatments of this topic include [28,29], which establish both the theoretical properties and practical implementations of nonparametric and semiparametric quantile regression procedures. In parallel, a growing body of work has addressed the more challenging setting in which the response remains scalar while the explanatory variable is functional. This framework, situated at the intersection of quantile regression and functional data analysis, poses substantial methodological and theoretical challenges due to the infinite-dimensional nature of the covariate. Notable contributions in this direction include [30,31,32,33], which develop estimation strategies and asymptotic results for conditional quantiles in functional regression contexts. It is important to emphasize, however, that the majority of these investigations operate under the assumption of fully observed samples. Such an assumption may be restrictive in many modern applications, where incomplete observations, measurement errors, or missing data mechanisms are frequently encountered, thereby motivating the need for more general inferential frameworks capable of accommodating these practical complexities.

In the context of missing data, Ref. [34] tackled the problem of nonparametric quantile regression estimation for the regression function operator in settings where the functional data exhibit responses that are missing at random in the univariate case. This extension takes into account the presence of missing data, which is a common occurrence in practical settings. Missing data present significant challenges in data analysis, as they complicate the process of estimating the underlying statistical models. It is crucial for practitioners to assess the validity of the assumptions underpinning their statistical models to determine the reliability of the method’s output. It is well understood by experienced practitioners that real-world datasets rarely adhere perfectly to theoretical assumptions, and they develop intuitive understanding, often supported by formal tests, regarding the severity of deviations from these assumptions. One of the most common and problematic discrepancies between real-world data and theoretical models is the presence of missing data. Missing data are ubiquitous in modern statistics and pose significant challenges across a wide range of applications. The issue of missing data is central in statistical theory, and the concept of data being missing at random (MAR) has become a foundational idea in this context. As noted in [35], missingness can arise for a variety of reasons. For instance, data may be collected from multiple sources, each of which measures a different subset of variables, leading to inconsistencies in the observed data. In healthcare, for example, routine data collected on patients may differ across clinics or hospitals, resulting in missing data for some variables. Other common causes of missing data include participants refusing to answer sensitive questions, uncontrollable factors, sensor failure, data censoring, and privacy concerns [36]. The study of missing data mechanisms has been an area of extensive research, with comprehensive overviews provided in [36,37]. According to [36], missing data can be categorized into three distinct mechanisms: missing completely at random (MCAR), missing at random (MAR), and not missing at random (NMAR). The MCAR mechanism occurs when the missingness is completely independent of both observed and unobserved variables. The MAR mechanism arises when the missingness is dependent on observed variables but not on the missing values themselves, whereas the NMAR mechanism occurs when the missingness is related to the unobserved data. While MCAR is the simplest mechanism to handle, it is also the most unrealistic in practice. MAR is more complex but often a more reasonable assumption in empirical settings. NMAR mechanisms, although seemingly more natural, present greater challenges in terms of modeling and estimation. For empirical settings, methods based on MAR assumptions have been found to provide more accurate predictions for missing values compared to methods that assume NMAR mechanisms [38]. This underscores the practical relevance of MAR in real-world datasets, where missing data are rarely missing completely at random. The study of missing data remains a critical and ongoing area of research, as missing data can severely affect the performance of statistical algorithms. In particular, missing data can make certain algorithms inoperable without appropriate modifications. Recent research has continued to refine our understanding of missing data mechanisms, highlighting the challenges in accurately characterizing these mechanisms. Notably, papers such as [39,40,41,42] have addressed the complexities surrounding missing data mechanisms and their implications for statistical modeling.

The extension of quantiles to multivariate settings has also garnered significant attention in the literature. Various approaches for multivariate quantiles have been proposed, including works by [43,44,45,46,47,48,49,50,51,52,53]. The concept of multivariate conditional quantiles was first introduced in [54] and later extended in [55] for the conditional median, as well as in [56,57,58] for multivariate conditional quantiles. For asymptotic properties in the i.i.d. case, see [59]. Functional data analysis, particularly in the context of infinite-dimensional covariates, gained prominence following the work of [60], with further contributions in [2,61].

The primary objective of this work is to establish the strong consistency of the conditional geometric quantile estimator and to determine its associated rate of convergence. In addition, we investigate the asymptotic normality of the estimator and construct corresponding confidence regions. Our methodological framework departs from the conventional reliance on strong mixing assumptions by instead postulating that the data are generated by a strictly stationary and ergodic process. This broader dependence structure yields a more flexible and analytically tractable setting, circumventing the technical intricacies commonly associated with strong mixing conditions and their numerous variants; see, for instance [62]. The present study further extends the scope of quantile regression to a setting in which the response variable Y is multivariate, while the predictor X takes values in an infinite-dimensional space, such as a functional covariate, potentially observed in the presence of missing data. By operating under ergodicity rather than strong mixing, we enlarge the admissible class of dependence structures and thus provide a substantive and novel contribution to the existing literature on regression with complex data types.

For the reader’s convenience, we briefly recall the definition of ergodicity and its relationship with mixing conditions. Let $\{X_n,n\in \mathbb{Z}\}$ be a strictly stationary sequence. Define the backward and forward sigma-fields by ${\mathcal{A}}_n=\sigma (X_k:k\le n)\mathrm{and}{\mathcal{B}}_m=\sigma (X_k:k\ge m).$ The sequence is said to be strongly mixing if, as $n\to \infty$, ${sup}_{A\in {\mathcal{A}}_0,B\in {\mathcal{B}}_n}|\mathbb{P}(A\cap B)-\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)|=\alpha \left(n\right)⟶0.$ By contrast, the sequence is ergodic if

$$\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\sum _{k=0}^{n-1}\left|\mathbb{P}\left(A\cap {\tau }^{-k}B\right)-\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)\right|=0,$$

where $\tau$ denotes the shift (time-evolution) transformation. It is worth noting that the notion of strong mixing adopted above is stronger than what is often termed strong mixing in the context of measure-preserving dynamical systems, where one typically requires only that ${lim}_{n\to \infty }\mathbb{P}(A\cap {\tau }^{-n}B)=\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)$ for all measurable sets A and B; see [63]. Consequently, strong mixing implies ergodicity, whereas the converse does not generally hold (see [64]). Several motivations for adopting ergodic dependence structures in place of mixing conditions are discussed in [65,66,67,68,69,70,71], where detailed definitions and illustrative examples of ergodic processes are also provided.

1.1. Paper Organization

The structure of the present paper is as follows. Section 2 introduces the statistical problem and the estimation procedures. Section 3 is devoted to presenting the core results of the study. In Section 3.2, we establish the uniform convergence rate for a general kernel estimator. Subsequently, Section 3.3 provides the asymptotic normality of the proposed estimator. In Section 4, the finite sample performance of the proposed method is investigated through simulations. In Section 5, we illustrate the method by means of an empirical application of the method. The paper concludes with Section 6, which summarizes the key findings and outlines potential directions for future research. For completeness, the mathematical foundations and detailed derivations are systematically presented in Section 7.

1.2. Notation

For the reader’s convenience, we summarize below the main notation and key quantities used throughout the paper, organized according to (i) the functional and probabilistic framework (Table 1), (ii) the geometric quantile model and associated population objects (Table 2), and (iii) the kernel-based estimators together with their asymptotic counterparts (Table 3).

Table 1. Functional and probabilistic framework.

Symbol	Meaning
(X,Y)	Random element taking values in $E\times {\mathbb{R}}^q$
E	Semi-metric functional covariate space (e.g., Hilbert or Banach space)
d(x,$x^{\prime }$)	Semi-metric measuring distance between curves in E
$⟨·,·⟩$	Euclidean inner product in ${\mathbb{R}}^q$
$\parallel ·\parallel$	Euclidean norm in ${\mathbb{R}}^q$
$B^q$	Unit ball $\{u\in {\mathbb{R}}^q:\parallel u\parallel <1\}$
${(X_i,Y_i)}_{i\ge 1}$	Strictly stationary ergodic sequence of observations
${\mathcal{F}}_i$	Sigma-field generated by past observations up to time i
${\mathcal{A}}_i$	Sigma-field generated by past observations and $X_{i+1}$
$\tau$	Shift operator of the underlying stochastic process
$F_x\left(h\right)$	Small-ball probability $\mathbb{P}\left(d\right(X,x)\le h)$
${F_x}^{{\mathcal{F}}_i}$(h)	Conditional small-ball probability given ${\mathcal{F}}_i$
$\varphi$(h)	Rate function such that $F_x\left(h\right)\sim \varphi \left(h\right)f_1\left(x\right)$ as $h\to 0$
$f_1\left(x\right)$	Leading term in the small-ball expansion
${\tau }_0$(s)	Limit ratio $\varphi \left(hs\right)/\varphi \left(h\right)$ as $h\to 0$

Table 2. Geometric quantile model and population quantities.

$\mathbf{Symbol}$	Meaning
u	Direction parameter in $B^q$ defining the geometric quantile
m	Candidate location vector in ${\mathbb{R}}^q$
$\mathcal{R}(u;\theta )$	Geometric quantile loss $\parallel \theta \parallel +⟨u,\theta ⟩$
${\mathcal{G}}^{u,x}\left(m\right)$	Conditional risk $\mathbb{E}\left[\mathcal{R}\right(u;Y-m)\mid X=x]$
${\mathbb{Q}}^{u,x}$	Conditional geometric quantile $arg{min}_m{\mathcal{G}}^{u,x}\left(m\right)$
${\nabla }_m{\mathcal{G}}^{u,x}\left(m\right)$	Gradient of the conditional risk with respect to m
${\mathbb{M}}^x\left(m\right)$	Hessian matrix of ${\mathcal{G}}^{u,x}\left(m\right)$
$\mathbb{D}(u;\theta )$	Subgradient of $\mathcal{R}$: $\theta /\parallel \theta \parallel +u$
$\mathbb{L}(y,m)$	Local curvature matrix of the geometric loss
${\Lambda }^x\left(m\right)$	Conditional covariance matrix of $\mathbb{D}(u;Y-m)$ given $X=x$

Table 3. Kernel estimators and asymptotic quantities.

$\mathbf{Symbol}$	Meaning
$K\left(·\right)$	Kernel function supported on $[0,1]$
$h=h_n$	Bandwidth parameter with $h\to 0$ as $n\to \infty$
${\mathcal{K}}_i\left(x\right)$	Kernel weight $K(d(x,X_i)/h)$
$w_{n,i,0}\left(x\right)$	Nadaraya–Watson weight (complete data)
$w_{n,i,1}\left(x\right)$	MAR-corrected Nadaraya–Watson weight
${\mathcal{G}}_n^{u,x}\left(m\right)$	Kernel estimator of ${\mathcal{G}}^{u,x}\left(m\right)$
${\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)$	MAR-weighted kernel estimator of ${\mathcal{G}}^{u,x}\left(m\right)$
${\mathbb{Q}}_n^{u,x}$	Kernel estimator of ${\mathbb{Q}}^{u,x}$ (complete data)
${\widehat{\mathbb{Q}}}_n^{u,x}$	Kernel estimator under MAR
${\varrho }_i$	Response indicator (1 if $Y_i$ observed)
$p\left(x\right)$	Response probability $\mathbb{P}(\varrho =1\mid X=x)$
$p_n\left(x\right)$	Kernel estimator of $p\left(x\right)$
$F_{n,x}\left(h\right)$	Empirical small-ball probability
${\mathcal{M}}_1,{\mathcal{M}}_2$	Kernel small-ball constants in asymptotic variance
${\mathbb{V}}^x\left({\mathbb{Q}}^{u,x}\right)$	Asymptotic covariance matrix of ${\widehat{\mathbb{Q}}}_n^{u,x}$

2. Statistical Framework

Let $q\ge 2$ be a given integer, and let $(X,Y)$ be a random element taking values in $E\times {\mathbb{R}}^q$, where E is a semi-metric space equipped with a semi-metric $d(·,·)$ (e.g., a Hilbert or Banach space). Denote by $⟨·,·⟩$ the standard inner product on ${\mathbb{R}}^q$, and by $\parallel ·\parallel$ the associated norm. Define the unit ball of ${\mathbb{R}}^q$ as $B^q:=\{z\in {\mathbb{R}}^q: \parallel z\parallel <1\}$. Let $\mathcal{R}:B^q\times {\mathbb{R}}^q\to \mathbb{R}$ be a loss function, defined for each pair $(u,\theta )\in B^q\times {\mathbb{R}}^q$ by

$$\mathcal{R}(u;\theta ):= \parallel \theta \parallel + ⟨u,\theta ⟩.$$

(1)

For any $(x,u)\in E\times B^q$ and $m\in {\mathbb{R}}^q$, define the conditional loss function as

$${\mathcal{G}}^{u,x}\left(m\right):=\mathbb{E}[\mathcal{R}(u;Y-m)\mid X=x].$$

(2)

The conditional geometric quantile is then defined, for a fixed $x\in E$ and $u\in B^q$, as the minimizer of ${\mathcal{G}}^{u,x}\left(m\right)$:

$${\mathbb{Q}}^{u,x}:={\mathbb{Q}}^u\left(x\right):=arg\underset{m\in {\mathbb{R}}^q}{min}{\mathcal{G}}^{u,x}\left(m\right)=arg\underset{m\in {\mathbb{R}}^q}{min}{\int }_{{\mathbb{R}}^q}\mathcal{R}(u;y-m)dF(y\mid x),$$

(3)

where $F(·\mid x)$ is the conditional multivariate cumulative distribution function of Y given $X=x$. Suppose, in the case of complete data, that we observe a sequence ${\left\{(X_i,Y_i)\right\}}_{i\ge 1}$ of copies of $(X,Y)$, assumed to be strictly stationary and ergodic. The Nadaraya–Watson-type estimator of ${\mathbb{Q}}^{u,x}$, denoted ${\mathbb{Q}}_n^{u,x}:={\mathbb{Q}}_n^u\left(x\right)$, is based on the sample ${\left\{(X_i,Y_i)\right\}}_{i=1}^n$. It is defined as the minimizer of a kernel-based empirical approximation of ${\mathcal{G}}^{u,x}\left(m\right)$:

$${\mathbb{Q}}_n^{u,x}:=arg\underset{m\in {\mathbb{R}}^q}{min}{\mathcal{G}}_n^{u,x}\left(m\right),$$

where

$${\mathcal{G}}_n^{u,x}\left(m\right):={\displaystyle \frac{{\sum }_{i=1}^n{\mathcal{K}}_i\left(x\right)\mathcal{R}(u;Y_i-m)}{{\sum }_{i=1}^n{\mathcal{K}}_i\left(x\right)}}={\displaystyle \frac{{\mathcal{G}}_{n,2}^{u,x}\left(m\right)}{{\mathcal{G}}_{n,1}^x}}.$$

Here, ${\mathcal{K}}_i\left(x\right):=K(d(x,X_i)/h)$, with $K\left(·\right)$ being a kernel function and $h:=h_n$ a bandwidth parameter decreasing to zero as $n\to \infty$. Additionally, the terms ${\mathcal{G}}_{n,j}^{u,x}\left((j-1)m\right)$, for $j=1,2$, are defined as

$${\mathcal{G}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)={\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n{\mathcal{R}}^{j-1}\left(u;Y_i-m\right){\mathcal{K}}_i\left(x\right),\mathrm{for}j=1,2,$$

(4)

with

$${\mathcal{G}}_{n,1}^x\left(0\right):={\mathcal{G}}_{n,1}^x.$$

In the missing data mechanism with MAR for the response variable, the available incomplete sample of size n from $(X,Y,\varrho )$ is given by

$$\left\{\left(X_i,Y_i,{\varrho }_i\right),1\le i\le n\right\},$$

where $X_i$ is observed completely, ${\varrho }_i=1$ if $Y_i$ is observed, and ${\varrho }_i=0$ otherwise. Meanwhile, the Bernoulli random variable $\varrho$ satisfies

$$\mathbb{P}(\varrho =1\mid X=x,Y=y)=\mathbb{P}(\varrho =1\mid X=x)=p\left(x\right),$$

where $p\left(x\right)$ is a function, called the conditional probability of observing the response given the predictor, which is often unknown. This mechanism shows that $\varrho$ and Y are conditionally independent given X. Therefore, the estimator is

$${\widehat{\mathbb{Q}}}_n^{u,x}:=arg\underset{m\in {\mathbb{R}}^q}{min}{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right),$$

where

$${\widehat{\mathcal{G}}}_n^{u,x}\left(m\right):={\displaystyle \frac{{\sum }_{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m\right)}{{\sum }_{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)}}:={\displaystyle \frac{{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\widehat{\mathcal{G}}}_{n,1}^x}},$$

when the denominator is not zero, and ${\mathcal{K}}_i\left(x\right)=K\left({\displaystyle \frac{d(x,X_i)}{h}}\right)$. The terms ${\widehat{\mathcal{G}}}_{n,j}^{u,x}\left(m\right)$ are defined as

$${\widehat{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)={\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n{\varrho }_i{\mathcal{R}}^{j-1}\left(u;Y_i-m\right){\mathcal{K}}_i\left(x\right),\mathrm{for}j=1,2,$$

(5)

with

$${\widehat{\mathcal{G}}}_{n,1}^x\left(0\right):={\widehat{\mathcal{G}}}_{n,1}^x.$$

We now introduce some definitions and notation for later use. Denote by $B^{\top }$ the transpose of the matrix B, and let ${\parallel B\parallel }^2=tr\left(B^{\top }B\right)$ be the norm trace. Notice that for any $(u,\theta )\in B^q\times {\mathbb{R}}^q$, the function $\mathcal{R}:(u,\theta )\mapsto \mathcal{R}(u;\theta )$ is differentiable with respect to $\theta$, everywhere except at $0_{{\mathbb{R}}^q}:=0$. Its derivative, defined by continuity extension, is given by

$$\mathbb{D}(u;\theta )=\mathcal{D}\left(\theta \right)+u:={\displaystyle \frac{\theta }{\parallel \theta \parallel }}+u\mathrm{when}\theta \ne 0,$$

and $\mathcal{D}\left(\theta \right):={\displaystyle \frac{\theta }{\parallel \theta \parallel }}=0$ when $\theta =0$. For any $y\ne m$, we define

$$\mathbb{L}(y,m)={\displaystyle \frac{1}{\parallel y-m\parallel }}\left({\mathbf{I}}_q-\mathcal{D}(y-m){\mathcal{D}}^{\top }(y-m)\right),$$

where ${\mathbf{I}}_q$ is the $q\times q$ identity matrix. We denote by ${\nabla }_m{\mathcal{G}}^{u,x}\left(m\right)$ the gradient of the function ${\mathcal{G}}^{u,x}\left(m\right)$ and by ${\mathbb{M}}^x\left(m\right)$ its Hessian matrix with respect to m. According to [44,46], we have

$${\nabla }_m{\mathcal{G}}^{u,x}\left(m\right)=-\mathbb{E}\left[\mathbb{D}(u;Y-m)\mid X=x\right],$$

(6)

and

$${\mathbb{M}}^x\left(m\right)=\mathbb{E}\left[\mathbb{L}(Y,m)\mid X=x\right].$$

Notice that ${\mathbb{M}}^x\left(m\right)$ is bounded whenever $\mathbb{E}\left[{\parallel Y-m\parallel }^{-1}\mid X=x\right]<\infty$. From (3) and (6), it follows that the conditional geometric quantile ${\mathbb{Q}}^{u,x}$ may be implicitly defined as a zero with respect to m of the following equation:

$${\nabla }_m{\mathcal{G}}^{u,x}\left(m\right)=0,$$

and its estimator ${\mathbb{Q}}_n^{u,x}$ may be viewed as a zero with respect to m of the equation ${\nabla }_m{\mathcal{G}}_n^{u,x}\left(m\right)=0$, that is,

$${\nabla }_m{\mathcal{G}}_n^{u,x}\left({\mathbb{Q}}_n^{u,x}\right)=0,$$

with

$${\nabla }_m{\mathcal{G}}_n^{u,x}\left(m\right)=-\sum _{i=1}^n{w}_{n,i,0}\left(x\right)\mathbb{D}\left(u;Y_i-m\right),m\in {\mathbb{R}}^q,x\in E,\mathrm{and}u\in B^q.$$

Similarly, the estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$, based on the functional stationary ergodic data with MAR, may be viewed as a zero with respect to m of the equation ${\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)=0$, that is,

$${\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)=0,$$

(7)

with

$${\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)=-\sum _{i=1}^n{w}_{n,i,1}\left(x\right)\mathbb{D}\left(u;Y_i-m\right),m\in {\mathbb{R}}^q,x\in E,\mathrm{and}u\in B^q,$$

where $w_{n,i,j}\left(x\right):={\displaystyle \frac{{\varrho }_i^j{\mathcal{K}}_i\left(x\right)}{{\sum }_{i=1}^n{\varrho }_i^j{\mathcal{K}}_i\left(x\right)}}$ for $j=0,1$ are the so-called Nadaraya–Watson weights. To state the asymptotic results, some further notation are required. Let

$${\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)={\displaystyle \frac{{\sum }_{i=1}^n{\varrho }_i\mathcal{R}\left(u;Y_i-m\right){\mathcal{K}}_i\left(x\right)}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}$$

and

$${\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)=-{\displaystyle \frac{{\sum }_{i=1}^n{\varrho }_i\mathbb{D}\left(u;Y_i-m\right){\mathcal{K}}_i\left(x\right)}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}.$$

It follows from (7) that

$${\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)=-{\displaystyle \frac{{\sum }_{i=1}^n{\varrho }_i\mathbb{D}\left(u;Y_i-{\widehat{\mathbb{Q}}}_n^{u,x}\right){\mathcal{K}}_i\left(x\right)}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}=0.$$

Thus, we can write

$${\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)-{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)=-{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right).$$

(8)

For each $j\in \{1,\dots ,q\}$, Taylor’s expansion of the real-valued function ${\displaystyle \frac{\partial {\widehat{\mathcal{G}}}_n^{u,x}}{\partial m_j}}$ at the point ${\mathbb{Q}}^{u,x}$ implies the existence of a random vector ${\xi }_n\left(j\right)={\left({\xi }_{n,1}\left(j\right),\dots ,{\xi }_{n,q}\left(j\right)\right)}^{\top }$ such that

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\widehat{\mathcal{G}}}_n^{u,x}}{\partial m_j}}\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)-{\displaystyle \frac{\partial {\widehat{\mathcal{G}}}_n^{u,x}}{\partial m_j}}\left({\mathbb{Q}}^{u,x}\right)=& \sum _{k=1}^q{\displaystyle \frac{{\partial }^2{\widehat{\mathcal{G}}}_n^{u,x}}{\partial m_j\partial m_k}}\left({\xi }_n\left(j\right)\right)\left({\widehat{\mathbb{Q}}}_{n,k}^{u,x}-{\mathbb{Q}}_k^{u,x}\right),\hfill \\ \hfill \left|{\xi }_{n,k}\left(j\right)-{\mathbb{Q}}_k^{u,x}\right|\le & \left|{\widehat{\mathbb{Q}}}_{n,k}^{u,x}\left(j\right)-{\mathbb{Q}}_k^{u,x}\right|.\hfill \end{array}$$

Define the $q\times q$ matrix ${\widehat{\mathbb{M}}}_n^x\left({\xi }_n\left(j\right)\right)={\left({\widehat{\mathbb{M}}}_{n,k,j}^x\left({\xi }_n\left(j\right)\right)\right)}_{1\le k,j\le q}$ by setting

$${\widehat{\mathbb{M}}}_{n,k,j}^x\left({\xi }_n\left(j\right)\right)={\displaystyle \frac{{\partial }^2{\widehat{\mathcal{G}}}_n^{u,x}}{\partial m_j\partial m_k}}\left({\xi }_n\left(j\right)\right),$$

where for all $m\in {\mathbb{R}}^q$ and $x\in E$,

$${\widehat{\mathbb{M}}}_{n,k,j}^x\left(m\right)=\sum _{i=1}^n{\mathbb{L}}_{k,j}(Y_i,m)\times {\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}},$$

with ${\mathbb{L}}_{k,j}(Y_i,m)={\displaystyle \frac{1}{\parallel Y_i-m\parallel }}\left[{\delta }_{k,j}-{\mathcal{D}}_k(Y_i-m){\mathcal{D}}_j(Y_i-m)\right]$. This equation can be rewritten as

$${\widehat{\mathbb{M}}}_n^x\left({\xi }_n\left(j\right)\right)\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right)=-{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right).$$

(9)

This equation will play a key role in determining the conditional bias and the rate of strong consistency of the conditional quantile estimator.

The Algorithm 1 summarizes the computation of ${\widehat{\mathbb{Q}}}_n^{u,x}$ in the MAR setting.

Algorithm 1 Computation of ${\widehat{\mathbb{Q}}}_n^{u,x}$ (MAR kernel geometric quantile)

3. Main Results

3.1. Assumptions

In our analysis, the following assumptions and notations are necessary to study the asymptotic properties of the estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$ for the conditional geometric quantile ${\mathbb{Q}}^{u,x}$. Let ${\mathcal{F}}_i$ be the $\sigma$-field generated by the pairs $\left(\left(X_1,Y_1\right),\dots ,\left(X_i,Y_i\right)\right)$, and let ${\mathcal{A}}_i$ be the $\sigma$-field generated by $\left(\left(X_1,Y_1\right),\dots ,\left(X_i,Y_i\right),X_{i+1}\right)$. Let $B(x,h)$ be a ball centered at $x\in E$ with radius h, and let $D_i\left(x\right):=d\left(x,X_i\right)$, where $D_i\left(x\right)$ is a non-negative real-valued random variable. Working on the probability space $(\Omega ,\mathcal{A},\mathbb{P})$, define the distribution function of the random variable $D_i\left(x\right)$ as

$$F_x\left(h\right):=\mathbb{P}\left(D_i\left(x\right)\le h\right)=\mathbb{P}\left(X_i\in B(x,h)\right),$$

and the conditional distribution function given the $\sigma$-field ${\mathcal{F}}_i$ as

$$F_x^{{\mathcal{F}}_i}\left(h\right):=\mathbb{P}\left(X_i\in B(x,h)\mid {\mathcal{F}}_i\right).$$

Denote by $o_{\mathrm{a}.\mathrm{s}.}\left(h\right)$ a real random function ℓ such that $\mathsf{\ell }\left(h\right)/h$ converges to zero almost surely as $h\to 0$. Similarly, define ${\mathcal{O}}_{\mathrm{a}.\mathrm{s}.}\left(h\right)$ as a real random function ℓ such that $\mathsf{\ell }\left(h\right)/h$ is almost surely bounded. Let $x\in E$, and denote by ${\mathcal{V}}_x$ a neighborhood of x. Our results are stated under the following assumptions, which are gathered here for easy reference:

(A1)

$K\left(·\right)$ is a non-negative, bounded kernel of class ${\mathcal{C}}^1$ over its support $[0,1]$, such that $K\left(1\right)>0$. The derivative $K^{\prime }\left(·\right)$ exists on $[0,1]$ and satisfies the condition $K^{\prime }\left(t\right)<0$ for all $t\in [0,1]$, and $\left|{\int }_0^1{\left(K^j\right)}^{\prime }\left(t\right)dt\right|<\infty$ for $j=1,2$.

(A2)

For $x\in E$, there exists a sequence of non-negative random functionals ${\left(f_{i,1}\right)}_{i\ge 1}$ almost surely bounded by a sequence of deterministic quantities ${\left(b_i\left(x\right)\right)}_{i\ge 1}$, a sequence of random functions ${\left(g_{i,x}\right)}_{i\ge 1}$, a deterministic non-negative bounded functional $f_1$, and a non-negative real function $\varphi$ tending to zero as its argument tends to 0, such that

(i)

$F_x\left(h\right)=\varphi \left(h\right)f_1\left(x\right)+o\left(\varphi \left(h\right)\right)$ as $h\to 0$.

(ii)

For any $i\in \mathbb{N}$, $F_x^{{\mathcal{F}}_{i-1}}\left(h\right)=\varphi \left(h\right)f_{i,1}\left(x\right)+g_{i,x}\left(h\right)$, with $g_{i,x}\left(h\right)=o_{\mathrm{a}.\mathrm{s}.}\left(\varphi \left(h\right)\right)$ as $h\to 0$, and $g_{i,x}\left(h\right)/\varphi \left(h\right)$ almost surely bounded. Moreover, $n^{-1}{\sum }_{i=1}^n{g}_{i,x}^j\left(h\right)=o_{\mathrm{a}.\mathrm{s}.}\left({\varphi }^j\left(h\right)\right)$ as $n\to \infty$, for $j=1,2$.

(iii)

$n^{-1}{\sum }_{i=1}^n{f}_{i,1}^j\left(x\right)\to f_1^j\left(x\right)$ almost surely as $n\to \infty$, for $j=1,2$.

(iv)

There exists a nondecreasing bounded function ${\tau }_0$ such that, uniformly in $s\in [0,1]$, ${\displaystyle \frac{\varphi \left(hs\right)}{\varphi \left(h\right)}}={\tau }_0\left(s\right)+o\left(1\right)$ as $h\downarrow 0$, and for $j\ge 1$, ${\int }_0^1{\left(K^j\left(t\right)\right)}^{\prime }{\tau }_0\left(t\right)dt<\infty$.

(v)

$n^{-1}{\sum }_{i=1}^n{b}_i\left(x\right)\to D\left(x\right)<\infty$ as $n\to \infty$.

(A3)

(i) For all $x\in E$, $(e,u)\in {\mathbb{R}}^q\times B^q$, we have

• $$\left|{\left(e^{\top }\right)}^j{\partial }_m^{\left(j\right)}{\mathcal{G}}^{u,x}\left(m\right)-{\left(e^{\top }\right)}^j{\partial }_m^{\left(j\right)}{\mathcal{G}}^{u,x^{\prime }}\left(m^{\prime }\right)\right|\le c_1\left(d^{\beta }\left(x,x^{\prime }\right)+{\parallel m-m^{\prime }\parallel }^{\delta }\right)$$

  uniformly in m ($j\in \{0,1\}$), for some $\beta >0,\delta >1$ and constant $c_1>0$, whenever $x^{\prime }\in {\mathcal{V}}_x$. By convention, ${\left(e^{\top }\right)}^0=1$.

(ii)

For $(s,m)\in \mathbb{R}\times {\mathbb{R}}^q$ denote by ${\mathcal{Z}}_s^x\left(m\right):=\mathbb{E}\left({∥Y-m∥}^s\mid X=x\right)$, for $x\in E$,

$$\mathbb{E}\left({∥Y-m∥}^s\mid {\mathcal{A}}_{i-1}\right)=\mathbb{E}\left({∥Y-m∥}^s\mid X_i\right)={\mathcal{Z}}_s^{X_i}\left(m\right).$$

The function ${\mathcal{Z}}_s^x\left(m\right)$ is continuous at x and ${\mathcal{Z}}_s^x\left({\mathbb{Q}}^{u,x}\right)<\infty$, for $s\ge 2$.

(A4)

For all $u\in B^q$ and $m\in {\mathbb{R}}^q$, the conditional mean of $\mathcal{R}(u,Y_i-m)$ and $\mathbb{D}(u,Y_i-m)$ given ${\mathcal{A}}_{i-1}$ depend only on $X_i$, i.e., for any $i\ge 1$:

(i)

$\mathbb{E}\left(\mathcal{R}(u,Y_i-m)\mid {\mathcal{A}}_{i-1}\right)={\mathcal{G}}^{u,X_i}\left(m\right)$ a.s.

(ii)

$\mathbb{E}\left[\mathbb{D}(u,Y_i-m)\mid {\mathcal{A}}_{i-1}\right]=\mathbb{E}\left[\mathbb{D}(u,Y_i-m)\mid X_i\right]:=-{\nabla }_m{\mathcal{G}}^{u,X_i}\left(m\right)$ a.s.

(A5)

For $m\ge 1$ and any $(k,j)$, $1\le k,j\le q$, we have

$$\mathbb{E}\left[{\left|{\mathbb{L}}_{k,j}\left(Y,{\mathbb{Q}}^{u,x}\right)\right|}^m\mid {\mathcal{A}}_{i-1}\right]=\mathbb{E}\left[{\left|{\mathbb{L}}_{k,j}\left(Y,{\mathbb{Q}}^{u,x}\right)\right|}^m\mid X_i\right]=:W_m\left(X_i,{\mathbb{Q}}^{u,x}\right)<\infty \mathrm{a}.\mathrm{s}.,$$

and

$$\underset{x^{\prime }:d\left(x,x^{\prime }\right)\le h}{sup}\left|W_m\left(x^{\prime },{\mathbb{Q}}^{u,x}\right)-W_m\left(x,{\mathbb{Q}}^{u,x}\right)\right|=o\left(1\right).$$

(A6)

${sup}_{x^{\prime }\in B(x,h)}∥{\mathbb{M}}^{x^{\prime }}\left({\mathbb{Q}}^{u,x}\right)-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥=o\left(1\right)$ a.s.

(A7)

For $k\in \{1,\dots ,q\}$, we have

$$\underset{x^{\prime }\in B(x,h)}{sup}\left|{\displaystyle \frac{\partial {\mathcal{G}}^{u,x^{\prime }}\left({\mathbb{Q}}^{u,x}\right)}{\partial m_k}}-{\displaystyle \frac{\partial {\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)}{\partial m_k}}\right|=o\left(1\right)\mathrm{a}.\mathrm{s}.$$

(A8)

For some $\eta \ge 1$ and for any $(e,m)\in {\mathbb{R}}^{2q}$, $j\in \{0,1\}$, the real function

$${\Phi }_{\eta }^{x^{\prime }}\left(m\right):=\mathbb{E}\left[{\left|{\left(e^{\top }\right)}^j\mathbb{D}(u,Y-m)\right|}^{\eta }\mid X=x^{\prime }\right]$$

is continuous in ${\mathcal{V}}_x$.

(A9)

$p\left(·\right)$ is continuous in a neighborhood of x, that is

$$\underset{\{u:d(x,u)\le h\}}{sup}|p\left(u\right)-p\left(x\right)|=o\left(1\right)\mathrm{as}h\to 0.$$

Comments

As discussed by [2], the conditions (A1) and (A3)(i) are standard assumptions placed on the kernel function in the context of nonparametric FDA. Specifically, condition (A1) relates to the choice of the kernel function K, which plays a crucial role in nonparametric functional estimation. The selection of an appropriate kernel is a fundamental part of this approach. Notably, the Parzen symmetric kernel, a popular choice in many settings, is not suitable in this context. This is due to the fact that the random process $D_i=d(x,X_i)$ is non-negative, and hence, the kernel must be chosen with support on the interval $[0,1]$, ensuring its suitability for modeling such processes. This condition can be viewed as a natural generalization of the typical assumption made on kernels in the multivariate setting, where kernels are generally assumed to be spherically symmetric density functions. Moreover, the conditions $K\left(1\right)>0$ and $K^{\prime }<0$ are introduced to ensure that the moment $M_1$ remains positive for all limiting functions ${\tau }_0$. In cases involving non-smooth processes, such as when ${\tau }_0$ takes the form of the Dirac delta function at 1, the condition $K\left(1\right)>0$ is necessary to properly define the moments $M_j$, which are, in such cases, determined by the specific value of $K\left(1\right)$.

Condition (A2) demonstrates the ergodic nature of the data and introduces the small ball techniques used throughout this paper. These techniques are essential in deriving the results related to the behavior of the data over large scales, which is a central aspect of the analysis in FDA. Assumptions (A4) and (A3)(ii) further provide moment conditions that clarify the Markovian nature of the functional stationary ergodic data. These assumptions reinforce the structure of the data and are critical for deriving consistency and other asymptotic properties in the model.

Finally, conditions (A5)–(A8) introduce local continuity assumptions that are essential for establishing the main results of the paper. These conditions ensure that the processes involved in the analysis are sufficiently smooth for the theoretical results to hold. Without these assumptions, the analysis would lose much of its rigor. In addition, condition (A9) provides further continuity conditions that ensure the clarity and conciseness of the results. These conditions are carefully designed to support the main findings and contribute to the overall structure of the paper, making the results not only technically sound but also accessible and comprehensible.

3.2. Strong Consistency with Rate

Proposition 1.

Assume that conditions (A1)–(A4)(i) hold true and that

$$\begin{array}{c}n\varphi \left(h\right)\to \infty ,{\displaystyle \frac{logn}{n\varphi \left(h\right)}}\to 0\mathit{and}{\displaystyle \frac{n\varphi \left(h\right)h^{2\beta }}{logn}}\to 0\mathit{as}n\to \infty ,\mathit{where}\beta \mathit{is}\mathit{given}\mathit{in}\left(\mathrm{A}3\right)\left(\mathrm{i}\right),\end{array}$$

(10)

$$\begin{array}{c}{\displaystyle \underset{\parallel m\parallel \to \infty }{\overline{lim}}\parallel m\parallel \left|{\mathcal{G}}^{u,x}\left(m\right)\right|<\infty .}\end{array}$$

(11)

Then, for any fixed $x\in E$ and for a directional vector $u\in B^q$, we have

$$\underset{m\in {\mathbb{R}}^q}{sup}\left|{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right)\right|={\mathcal{O}}_{a.s.}\left(h^{\beta }\right)+{\mathcal{O}}_{a.s.}\left(\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right).$$

Theorem 1.

Under the same assumptions of Proposition 1, we get, for any $x\in E$ and $u\in B^q$,

$$\underset{n\to \infty }{lim}∥{\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}∥=0a.s.$$

(12)

In order to get the convergence rate of the above result, we first state and prove the following intermediate result, which deals with the consistence of the Hessian functional matrix.

Proposition 2.

Under assumptions (A1)–(A2), (A3)(i) (with $j=0$), (A4)(i) and Condition (10), we have

$$∥{\widehat{\mathbb{M}}}_n^x\left({\xi }_n\left(j\right)\right)-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥=o_{a.s.}\left(1\right),\mathit{as}n\to \infty .$$

Using now Remark 4 and Lemma 5.3 of [44], we know that both matrix ${\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)$ itself and its inverse matrix exist whenever $q\ge 2$. It follows from this result combined with (9) and Proposition 2 that, for n large enough,

$${\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}=-{\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]}^{-1}{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)\mathrm{a}.\mathrm{s}.$$

(13)

The pointwise convergence rate of ${\widehat{\mathbb{Q}}}_n^{u,x}$ to ${\mathbb{Q}}^{u,x}$ is established under additional conditions in Theorem 2 below.

Theorem 2.

Under assumptions (A1)–(A3)(i) (with $j=0$), (A4)(i), (A5)–(A7) and condition (10), we have

$$∥{\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}∥={\mathcal{O}}_{a.s.}\left(\varphi \left(h\right)\right)+{\mathcal{O}}_{a.s.}\left(\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right).$$

3.3. Asymptotic Distribution

Let us now state the following theorem, which gives the weak convergence rate of the estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$ defined in (7). Below, we write $Z\stackrel{\mathcal{D}}{=}\mathcal{N}(\mu ,{\sigma }^2)$ whenever the random vector Z follows a normal law with expectation $\mu$ and covariance matrix ${\sigma }^2$, $\stackrel{\mathcal{D}}{\to }$ denotes the convergence in distribution and $\stackrel{\mathbb{P}}{\to }$ the convergence in probability.

Theorem 3.

Assume that Conditions (A1)–(A4), (A8) and (10) hold true. Then, we have

$$\sqrt{n\varphi \left(h\right)}\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right)\stackrel{\mathcal{D}}{⟶}{\mathcal{N}}_q\left(0,{\mathbb{V}}^x\left({\mathbb{Q}}^{u,x}\right)\right),$$

(14)

where

$${\mathbb{V}}^x\left({\mathbb{Q}}^{u,x}\right)={\displaystyle \frac{{\mathcal{M}}_2}{{\mathcal{M}}_1^2{f}_1\left(x\right)p\left(x\right)}}{\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]}^{-1}{\Lambda }^x\left({\mathbb{Q}}^{u,x}\right){\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]}^{-1},$$

and

$${\Lambda }^x\left({\mathbb{Q}}^{u,x}\right):=\mathbb{E}\left\{\mathbb{D}\left(u;Y-{\mathbb{Q}}^{u,x}\right){\mathbb{D}}^{\top }\left(u;Y-{\mathbb{Q}}^{u,x}\right)\mid X=x\right\}.$$

Since the asymptotic variance of the estimator given in Theorem 3 involves several unknown quantities, it is generally not feasible to use it directly for practical inference. In particular, it does not provide a straightforward way to construct confidence regions. The following corollary presents an alternative central limit theorem to address this limitation. This version enables the construction of an asymptotic $100(1-\alpha )\%$ (with $\alpha \in (0,1)$) conditional confidence region for ${\mathbb{Q}}^{u,x}$. Before stating the corollary, we introduce the following nonparametric estimators for ${\mathbb{V}}^x\left({\mathbb{Q}}^{u,x}\right)$ and ${\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)$. These estimators incorporate the sample observations and are designed to be implementable in practice, thus paving the way for effective inference based on observed data. Specifically, we define

$${\Lambda }_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)=\sum _{i=1}^n{w}_{n,i}\left(x\right)\mathbb{D}\left(u,Y_i-{\widehat{\mathbb{Q}}}_n^{u,x}\right){\mathbb{D}}^{\top }\left(u,Y_i-{\widehat{\mathbb{Q}}}_n^{u,x}\right)$$

and

$${\mathbb{M}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)=\sum _{i=1}^n{w}_{n,i}\left(x\right)\mathbb{L}\left(Y_i,{\widehat{\mathbb{Q}}}_n^{u,x}\right).$$

In addition, leveraging the decomposition of $F_x\left(\zeta \right)$ in (A2)(i), one can estimate ${\tau }_0\left(\zeta \right)$ by

$${\tau }_n\left(\zeta \right)={\displaystyle \frac{F_{n,x}\left(\zeta h\right)}{F_{n,x}\left(\zeta \right)}},\mathrm{where}{F}_{n,x}\left(\zeta \right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\mathbb{1}}_{\{d(x,X_i)\le \zeta \}}.$$

Subsequently, given a kernel $K\left(·\right)$, the terms ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ are replaced by their estimators ${\mathcal{M}}_{1,n}$ and ${\mathcal{M}}_{2,n}$, respectively, obtained by substituting ${\tau }_0$ with ${\tau }_n$ in their corresponding definitions. Thus, the estimator of $p\left(x\right)$ is defined by

$$p_n\left(x\right)={\displaystyle \frac{{\sum }_{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{\sum }_{i=1}^n{\mathcal{K}}_i\left(x\right)}}.$$

(15)

Corollary 1, presented below as a slight adaptation of Theorem 3, provides a more practical and useful formulation of our results.

Corollary 1.

Assume that the conditions stated in Theorem 3 hold and that $K^{\prime }$ and ${\left(K^2\right)}^{\prime }$ are integrable functions. Furthermore, suppose

$$n{F}_{n,x}\left(h\right)\to \infty \mathit{and}{h}^{\beta }{\left(n{F}_{n,x}\left(h\right)\right)}^{1/2}\to 0,\mathit{as}n\to \infty ,$$

where β is defined in assumption (A3). Then, for any $x\in E$ with $f_1\left(x\right)>0$, it follows that

$${\displaystyle \frac{{\mathcal{M}}_{1,n}}{\sqrt{{\mathcal{M}}_{2,n}}}}\sqrt{{\displaystyle \frac{n{F}_{n,x}\left(h\right)p_n\left(x\right)}{{\Lambda }_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)}}}{\mathbb{M}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right)\stackrel{\mathcal{D}}{\to }{\mathcal{N}}_q\left(0,{\mathbf{I}}_q\right).$$

3.4. Confidence Region

Since the terms in Corollary 1 are all observable or can be consistently estimated, this result facilitates the construction of a confidence region for ${\mathbb{Q}}^{u,x}$. Observe that:

$${\left({\widehat{\mathbb{Q}}}_n^{u,x}-Q^{u,x}\right)}^{\top }{\left[{\mathbb{V}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)\right]}^{-1}\left({\widehat{\mathbb{Q}}}_n^{u,x}-Q^{u,x}\right)\stackrel{\mathcal{D}}{\to }{\chi }_q^2,$$

where

$${\left[{\mathbb{V}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)\right]}^{-1}={\displaystyle \frac{{\mathcal{M}}_{1,n}^{2}n{F}_{n,x}\left(h\right)p_n\left(x\right)}{{\mathcal{M}}_{2,n}}}{\mathbb{M}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right){\left[{\Lambda }_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)\right]}^{-1}{\mathbb{M}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right).$$

Thus, the asymptotic $100(1-\alpha )\%$ conditional confidence region for ${\mathbb{Q}}^{u,x}$ can be expressed as

$${\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right)}^{\top }{\left[{\mathbb{V}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)\right]}^{-1}\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right)\le {\chi }_q^2\left(\alpha \right),$$

where ${\chi }_q^2\left(\alpha \right)$ denotes the $100(1-\alpha )$-th percentile of the chi-squared distribution with q degrees of freedom. By replacing the unknown asymptotic variance terms with their consistent nonparametric estimators and leveraging the refined central limit theorem in Corollary 1, we obtain a practical procedure for constructing valid asymptotic confidence regions.

Remark 1.

To illustrate assumptions (A4) and (A3)(ii) as in [72], we provide several examples that demonstrate how these conditions are satisfied in various contexts:

(i) 

Long-memory discrete-time processes: Let ${\left({ϵ}_t\right)}_{t\in \mathbb{Z}}$ denote a white noise process with variance ${\sigma }^2$, and let I and B represent the identity operator and the backshift operator, respectively. [73] proved (see Theorem 1, p. 55) that the k-factor Gegenbauer process

$$\prod _{i\le i\le k}{\left(I-2v_{i}B+B^2\right)}^{d_i}{X}_t={ϵ}_t,$$

where $0<d_i<1/2$ if $|{\nu }_i| < 1$, or $0<d_i<1/4$ if $|{\nu }_i| = 1$, for $i=1,\dots ,k$, is a long-memory, stationary, causal, and invertible process. This process has a moving average representation, expressed as

$$X_t=\sum _{j\ge 0}{\psi }_j(d,v){ϵ}_{t-j},$$

where

$$\sum _{j=0}^{\infty }{\psi }_j^2(d,v)<\infty ,$$

ensuring that the process is well-behaved in the asymptotic sense.

On the other hand, Ref. [74] demonstrated that if ${\left({ϵ}_t\right)}_{t\in \mathbb{Z}}$ is a Gaussian process, the above process is not strongly mixing. Nevertheless, the moving average representation confirms that the process is stationary, Gaussian, and ergodic. This example highlights the subtlety of mixing conditions and the importance of the moving average representation in understanding the long-term behavior of the process.

(ii) 

The stationary solution of the linear Markov $\mathit{AR}\left(1\right)$ process: Consider the process

$$X_i={\displaystyle \frac{1}{2}}{X}_{i-1}+{ϵ}_i,$$

where $\left({ϵ}_i\right)$ are independent symmetric Bernoulli random variables taking values −1 and 1. This process is not α-mixing, as shown by [64], because of the inherent dependence structure between consecutive values of $X_i$. However, despite the lack of strong mixing, the process $\left(X_i\right)$ is Markovian, stationary, and ergodic. This example demonstrates how a Markov process, even when it does not exhibit strong mixing, can still exhibit ergodicity, which is crucial for many statistical analyses in time series and functional data analysis.

(iii) 

A stationary process with an $\mathit{AR}\left(1\right)$ representation: Let $\left(u_i\right)$ be an i.i.d. sequence uniformly distributed on $\{1,\dots ,9\}$, and define the process

$$X_t:=\sum _{i=0}^{\infty }{10}^{-i-1}{u}_{t-i},$$

where $u_t,u_{t-1},\dots$ represent the decimal expansion of $X_t$. This process is stationary and can be expressed in the form of an $\mathit{AR}\left(1\right)$ process:

$$X_t={\displaystyle \frac{1}{10}}{X}_{t-1}+{\displaystyle \frac{1}{10}}{u}_t={\displaystyle \frac{1}{10}}{X}_{t-1}+{\displaystyle \frac{1}{2}}+{ϵ}_t,$$

where ${ϵ}_t={\displaystyle \frac{1}{10}}{u}_t-{\displaystyle \frac{1}{2}}$ is a strong white noise. While this process is not α-mixing (see [75], Example A.3, p. 349), it is ergodic. This example illustrates a situation where a process that is not strongly mixing can still exhibit the necessary statistical properties of ergodicity, making it suitable for applications in nonparametric FDA.

Remark 2.

The present work contributes to the literature at the intersection of multivariate quantiles, functional data analysis, and incomplete-data inference by developing a unified framework that accommodates infinite-dimensional covariates, weak dependence, and MAR missingness within a single geometric quantile paradigm. In contrast with the classical theory of spatial or directional quantiles, which is predominantly formulated for i.i.d. data with finite-dimensional predictors, our analysis operates in a semi-metric functional space where asymptotic behavior is governed by small-ball probabilities. This shift in geometric setting requires a nonstandard combination of local smoothing arguments, ergodic techniques, and directional convex analysis, and leads to convergence rates and limit distributions that explicitly reflect the local concentration of trajectories around the conditioning curve. A further distinguishing feature is the endogenous treatment of missing responses. Rather than viewing missingness as a preprocessing issue, the MAR mechanism is embedded directly into the estimator through response indicators and is carried throughout the asymptotic development. As a consequence, the effective information content, the bias–variance trade-off, and the limiting covariance structure all depend on the local response probability $p\left(x\right)$, yielding an inferential theory that remains valid under heterogeneous observation patterns. To our knowledge, such a fully MAR-robust asymptotic analysis has not previously been established for conditional multivariate quantiles in a functional framework. From an applied perspective, the methodology extends the scope of functional regression beyond moment-based summaries. Directional conditional geometric quantiles provide a robust, geometry-aware characterization of $Y\mid X=x$ that is sensitive to tail behavior, asymmetry, and directional heterogeneity—features that are typically invisible to conditional mean or covariance models. The accompanying confidence regions translate the underlying functional sparsity and response incompleteness into interpretable measures of uncertainty, thereby enabling distribution-level inference in complex FDA settings. Altogether, the results place conditional geometric quantiles on the same methodological footing as classical nonparametric functional estimators, while substantially enriching the descriptive and inferential toolbox available for multivariate responses.

Remark 3.

The robustness of the conditional geometric quantile estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$ stems from the intrinsic geometry of the loss function

$$\mathcal{R}(u;\theta )= \parallel \theta \parallel + ⟨u,\theta ⟩,$$

which is a directional extension of the multivariate spatial ($L_1$-type) criterion. In contrast with quadratic losses, $\mathcal{R}$ grows linearly in $\parallel \theta \parallel$, so that extreme responses affect the estimating equations only through their direction relative to m, via the score $\mathbb{D}(u;Y-m)={\displaystyle \frac{Y-m}{\parallel Y-m\parallel }}+u$. Hence, large-magnitude outliers have bounded influence on the estimator. This mechanism parallels the classical robustness of spatial medians and geometric quantiles and remains valid in the conditional framework since kernel weights enter multiplicatively and do not alter the linear growth of the loss.

It is noteworthy that, despite the geometric robustness of spatial quantiles, their formal robustness properties have received comparatively little attention in the literature. To the best of our knowledge, breakdown behavior has mainly been investigated for related spatial outlyingness measures ([76,77]), for which explicit representations make such analysis more tractable. In contrast, analogous breakdown point results for spatial quantiles themselves are substantially more involved and have only recently begun to be explored; see [78] for advanced developments in this direction. The present work therefore contributes to a still-developing line of research by bringing spatial-quantile robustness into a conditional, functional-data setting.

To substantiate these robustness properties empirically, the simulation study was extended with contamination scenarios. A proportion $\epsilon \in \{0.05,0.10,0.15\}$ of responses was replaced by heavy-tailed or high-leverage outliers generated from inflated-variance or shifted distributions, while the functional covariates were kept unchanged. The performance of the proposed estimator was compared with that of a local conditional mean estimator based on squared loss. Under contamination, the geometric quantile estimator maintained substantially smaller bias and RMSE, and its confidence regions remained stable, whereas the mean-based estimator deteriorated rapidly as ε increased. These results confirm that the bounded-influence structure of the geometric loss translates into tangible finite-sample robustness.

This robustness mechanism is conceptually related to modern robust regression approaches designed to mitigate the impact of outliers (e.g., [79]), but differs fundamentally in construction: robustness here arises from the geometry of the loss itself rather than from trimming or reweighting schemes applied to a quadratic criterion. In this sense, the proposed estimator may be viewed as a nonparametric functional-data analogue of bounded-influence regression procedures, with robustness built directly into the defining estimating functional.

4. Simulation Study

In this section we investigate the finite-sample behavior of the conditional geometric quantile estimator introduced in this paper, focusing on a purely simulated functional setting with MAR responses. All tuning choices (basis, sample sizes, grid size, bandwidth, kernel, optimization tolerances, regularization level, Monte Carlo size for the pseudo-“ground truth”, etc.) are made explicit and are motivated theoretically. Throughout, confidence regions are built at nominal level $1-\alpha =0.95$.

4.1. Data-Generating Mechanism

4.1.1. Functional Covariate

For each replication and for each sample size n, we generate an i.i.d. sample $\{X_i:i=1,\dots ,n\}$ of square-integrable random trajectories on the unit interval $[0,1]$. These trajectories take values in the separable Hilbert space $L^2[0,1]$ equipped with the usual inner product

$${⟨f,g⟩}_2={\int }_0^{1}f\left(t\right)g\left(t\right)dt,f,g\in L^2[0,1],$$

and norm

$${\parallel f\parallel }_2={⟨f,f⟩}_2^{1/2}.$$

We fix the following three-dimensional orthonormal basis of $L^2[0,1]$:

$${\psi }_1\left(t\right)=1,{\psi }_2\left(t\right)=\sqrt{2}sin \left(\pi t\right),{\psi }_3\left(t\right)=\sqrt{2}cos \left(\pi t\right),t\in [0,1].$$

Each trajectory $X_i$ is observed on an equidistant grid $t_1,\dots ,t_T$ with $T=41$ points given by

$$t_{\mathsf{\ell }}={\displaystyle \frac{\mathsf{\ell }-1}{T-1}},\mathsf{\ell }=1,\dots ,T.$$

We consider the truncated Karhunen–Loève [80] representation

$$X_i\left(t\right)=\sum _{k=1}^3{\xi }_{ik}{\psi }_k\left(t\right),i=1,\dots ,n,$$

where ${({\xi }_{i1},{\xi }_{i2},{\xi }_{i3})}^{\top }$ are independent across i and have independent Gaussian coordinates

$${\xi }_{ik}\sim \mathcal{N}(0,{\lambda }_k),({\lambda }_1,{\lambda }_2,{\lambda }_3)=(1,0.5,0.25).$$

This choice mimics a realistic functional principal component decomposition with decreasing energy and ensures that most of the variability is captured by the first mode, while the remaining modes generate non-trivial oscillations. Numerically, each $X_i$ is represented by the vector of evaluations

$$X_i=\left(X_i\left(t_1\right),\dots ,X_i\left(t_T\right)\right)\in {\mathbb{R}}^T,$$

and distances between curves $x,x^{\prime }\in {\mathbb{R}}^T$ are computed using the empirical $L^2$ metric

$$d(x,x^{\prime })={\left\{{\displaystyle \frac{1}{T}}\sum _{\mathsf{\ell }=1}^T{\left(x\left(t_{\mathsf{\ell }}\right)-x^{\prime }\left(t_{\mathsf{\ell }}\right)\right)}^2\right\}}^{1/2},$$

which corresponds to a Riemann sum approximation of the $L^2$ norm.

4.1.2. Conditional Distribution of Y

The response is two-dimensional and real-valued:

$$Y_i={(Y_{i1},Y_{i2})}^{\top }\in {\mathbb{R}}^2,i=1,\dots ,n.$$

The conditional distribution of $Y_i$ depends on $X_i$ only through a scalar single-index

$$Z_i={⟨X_i,\beta ⟩}_2\approx {\displaystyle \frac{1}{T}}\sum _{\mathsf{\ell }=1}^T{X}_i\left(t_{\mathsf{\ell }}\right)\beta \left(t_{\mathsf{\ell }}\right),$$

with slope function

$$\beta \left(t\right)={\psi }_1\left(t\right)+{\psi }_2\left(t\right),t\in [0,1].$$

This design is deliberately aligned with the first two eigenfunctions of X, which makes the index sizeable and avoids degenerate small-ball probabilities. Given $Z_i=z$, we impose an elliptical conditional model

$$Y_i\mid \{Z_i=z\}\sim \mu \left(z\right)+{\epsilon }_i,\mu \left(z\right)={\left(m_1\left(z\right),m_2\left(z\right)\right)}^{\top },$$

where the deterministic regression functions are

$$m_1\left(z\right)=z,m_2\left(z\right)=2z^2,$$

and ${\epsilon }_i$ is a centered bivariate Student elliptical noise with degrees of freedom $\nu =5$ and covariance matrix

$${\epsilon }_i\sim t_{\nu }(0,\Sigma ),\Sigma =\left(\begin{array}{cc}{\sigma }_1^2& \rho {\sigma }_1{\sigma }_2\\ \rho {\sigma }_1{\sigma }_2& {\sigma }_2^2\end{array}\right),{\sigma }_1={\sigma }_2=1,\rho =0.5.$$

Equivalently, one can write

$${\epsilon }_i={\displaystyle \frac{W_i}{\sqrt{S_i/\nu }}},W_i\sim {\mathcal{N}}_2(0,\Sigma ),S_i\sim {\chi }_{\nu }^2,W_i \perp S_i,$$

so that the conditional law of $Y_i$ given $Z_i=z$ is bivariate Student with location $\mu \left(z\right)$ and scale $\Sigma$. The combination of a linear component $m_1$ and a strongly non-linear component $m_2$ yields markedly different signal-to-noise ratios for the two coordinates. The heavy-tailed Student innovation (rather than a Gaussian one) stresses the robustness of geometric quantiles in the presence of outliers and inflates the probability of extreme events along random directions. To visualize the conditional law, we fix a reference curve

$$x_{\mathrm{eval}}\left(t\right)\equiv 0,t\in [0,1],$$

so that its (scalar) index is $Z_{\mathrm{eval}}={⟨x_{\mathrm{eval}},\beta ⟩}_2=0$, and approximate the corresponding conditional distribution $Y\mid X=x_{\mathrm{eval}}$ by Monte Carlo. Specifically, we generate $N_0=2\times {10}^4$ independent draws

$$Y^{\left(j\right)}\sim Y|X=x_{\mathrm{eval}},j=1,\dots ,N_0,$$

and approximate, for each direction $u\in {\mathbb{R}}^2$ with ${\parallel u\parallel }_2=1$, the corresponding true conditional geometric quantile ${\widehat{\mathbb{Q}}}_n^{u,x_{\mathrm{eval}}}$ by

$${\widehat{\mathbb{Q}}}_n^{u,x_{\mathrm{eval}}}\approx arg\underset{m\in {\mathbb{R}}^2}{min}{\displaystyle \frac{1}{N_0}}\sum _{j=1}^{N_0}\left(\parallel Y^{\left(j\right)}{-m\parallel }_2+⟨u,Y^{\left(j\right)}-m⟩\right).$$

We focus on the three directions

$$u_1={(1,0)}^{\top },u_2={(0,1)}^{\top },u_3={(1,1)}^{\top }/\sqrt{2}.$$

These directions have been chosen to isolate marginal behavior ($u_1,u_2$) and a diagonal compromise ($u_3$), which is particularly informative for elliptical models. The resulting bivariate cloud, together with the three true conditional geometric quantiles $Q^{u_k}\left(x_{\mathrm{eval}}\right)$, $k=1,2,3$, is displayed in Figure 1. As expected, the Student cloud is globally elliptical but exhibits anisotropic dispersion and visible tail thickness along these directions.

Figure 1. Conditional elliptical distribution of $Y\mid X=x_{\mathrm{eval}}$ (Student model with $\nu =5$) and true geometric quantiles $Q^{u_k}\left(x_{\mathrm{eval}}\right)$ for three directions $u_1=(1,0)$, $u_2=(0,1)$ and $u_3=(1,1)/\sqrt{2}$.

4.1.3. Missingness Mechanisms

Missing responses are generated according to a MAR mechanism that depends on the scalar index $Z_i$. For each i, we generate a missingness indicator

$${\rho }_i\sim \mathrm{Bernoulli}\left\{p\left(Z_i\right)\right\},p\left(z\right)={\mathrm{logit}}^{-1}({\alpha }_0+{\alpha }_{1}z)={\displaystyle \frac{1}{1+exp\{-({\alpha }_0+{\alpha }_{1}z)\}}},$$

and only observations with ${\rho }_i=1$ are retained in the analysis. We consider three mechanisms of increasing departure from MCAR:

$$\begin{array}{cc}\mathrm{MCAR}:\hfill & ({\alpha }_0,{\alpha }_1)=(\mathrm{logit}\left(0.7\right),0),\hfill \\ \mathrm{MAR}1:\hfill & ({\alpha }_0,{\alpha }_1)=(0,-0.5),\hfill \\ \mathrm{MAR}2:\hfill & ({\alpha }_0,{\alpha }_1)=(-0.5,-1.0).\hfill \end{array}$$

In the MCAR case, $Pr({\rho }_i=1\mid X_i)=0.7$ for all $X_i$, so missingness is independent of $X_i$ and $Y_i$. In MAR1 the response probability decreases moderately with $Z_i$, which slightly favours small values of the index and induces a mild distortion of the conditional distribution. MAR2 represents a much more informative missingness mechanism, where high values of $Z_i$ are severely undersampled. Importantly, all three mechanisms are MAR in the sense of

$${\rho }_i\perp Y_i\mid X_i,$$

that is, the conditional distribution of ${\rho }_i$ depends on $X_i$ only through $Z_i$ and does not depend on $Y_i$ itself.

4.1.4. Sample Sizes and Replications

We consider three sample sizes $n\in \{250,500,1000\},$ which correspond to realistic moderate-to-large functional datasets. Very small samples ($n<200$) lead to unstable local behavior under strong MAR and have been discarded, while substantially larger sample sizes would be computationally demanding because of the repeated optimization of the geometric loss and variance estimation. For each configuration $(n,\mathrm{mechanism})$ and each direction $u\in \{u_1,u_2,u_3\}$ we perform $N_{\mathrm{MC}}=100$ independent Monte Carlo replications. At each replication, the true conditional geometric quantile $Q^u\left(x_{\mathrm{eval}}\right)$ is re-approximated by the Monte Carlo procedure described above with $N_0=2\times {10}^4$ draws from the conditional bivariate Student law. This ensures that the Monte Carlo error on the ‘ground truth’ is negligible compared to the sampling variability at the level of n.

4.2. Estimator, Kernel and Bandwidth

4.2.1. Directional Geometric Loss

For a fixed direction $u\in {\mathbb{R}}^2$ with ${\parallel u\parallel }_2=1$, the directional geometric loss function for a candidate center $m\in {\mathbb{R}}^2$ and an observation $y\in {\mathbb{R}}^2$ is defined by

$$R_u(y-m)={\parallel y-m\parallel }_2+⟨u,y-m⟩.$$

The (population) conditional geometric quantile ${\widehat{\mathbb{Q}}}^{u,x}$ in direction u is defined as the (unique) minimizer of

$$m\mapsto \mathbb{E}\{R_u(Y-m)\mid X=x\}.$$

In the sample, the local estimator ${\widehat{\mathbb{Q}}}_n^{u,x_{\mathrm{eval}}}$ minimises the weighted empirical risk

$${\widehat{G}}_u(m;x_{\mathrm{eval}})=\sum _{i=1}^n{w}_i\left(x_{\mathrm{eval}}\right)R_u(Y_i-m),$$

where the weights $w_i\left(x_{\mathrm{eval}}\right)$ are defined below and incorporate both the MAR mechanism and the kernel localization.

4.2.2. Kernel and Weights

We adopt the quadratic kernel on $[0,1]$,

$$K\left(t\right)={(1-t)}^2{\mathbf{1}}_{[0,1]}\left(t\right),t\in \mathbb{R},$$

which is compactly supported and twice differentiable on $(0,1)$, with

$$K^{\prime }\left(t\right)=-2(1-t){\mathbf{1}}_{[0,1]}\left(t\right),K^{\prime \prime }\left(t\right)=2{\mathbf{1}}_{(0,1)}\left(t\right).$$

For a given bandwidth $h>0$ and target curve $x_{\mathrm{eval}}$, denote the $L^2$-distance between $X_i$ and $x_{\mathrm{eval}}$ by

$$d_i=d(X_i,x_{\mathrm{eval}}),i=1,\dots ,n.$$

The raw (unnormalized) weight is

$${\tilde{w}}_i={\rho }_{i}K\left(d_i/h\right),$$

and the normalised weight used for estimation is

$$w_i\left(x_{\mathrm{eval}}\right)={\displaystyle \frac{{\tilde{w}}_i}{{\sum }_{j=1}^n{\tilde{w}}_j}}={\displaystyle \frac{{\rho }_{i}K(d_i/h)}{{\sum }_{j=1}^n{\rho }_{j}K(d_j/h)}},i=1,\dots ,n,$$

so that $w_i\left(x_{\mathrm{eval}}\right)\ge 0$ for all i and ${\sum }_{i=1}^n{w}_i\left(x_{\mathrm{eval}}\right)=1$ whenever at least one ${\tilde{w}}_i$ is positive. The factor ${\rho }_i$ cancels the missing responses, and the kernel localization $K(d_i/h)$ restricts attention to a neighborhood of $x_{\mathrm{eval}}$. The empirical small-ball probability at $x_{\mathrm{eval}}$ is defined by

$$F_n^{x_{\mathrm{eval}}}\left(h\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n\mathbf{1}\{d_i\le h\}.$$

4.2.3. Bandwidth Selection

The local neighborhood of $x_{\mathrm{eval}}$ is controlled by a bandwidth h on the $L^2$ distance. In principle, one could estimate h by leave-one-out cross-validation minimizing the local predictive geometric loss. While such a scheme is implemented in our general code, we observed that, in the present functional setting, it tends to be overly variable and tends to select bandwidths that are too small when the MAR mechanism is strong. This leads to unstable weights and quasi-singular local information matrices. For the Monte Carlo study reported here we therefore adopt a deterministic, data-driven bandwidth defined by the empirical $20\%$ quantile of the distances:

$$h=Q_{0.2}\left(\{d_1,\dots ,d_n\}\right),$$

that is, the smallest z such that

$${\displaystyle \frac{1}{n}}\sum _{i=1}^n\mathbf{1}\{d_i\le z\}\ge 0.2.$$

This choice is scale-adaptive, invariant with respect to monotone transformations of the distances and ensures that roughly $20\%$ of the sample receives non-negligible weight. It also guarantees that the empirical small-ball probability $F_n^{x_{\mathrm{eval}}}\left(h\right)$ remains well away from zero, which is crucial for the stability of the estimated asymptotic variance.

4.2.4. Variance Estimation and Confidence Regions

The asymptotic variance of ${\widehat{\mathbb{Q}}}_n^{u,x_{\mathrm{eval}}}$ is estimated using a plug-in matrix estimator as described in Section 3, with all empirical quantities made explicit here. For each i with ${\rho }_i=1$ and $K(d_i/h)>0$, define the directional gradient vector

$$D_i={\nabla }_m{R}_u(Y_i-{\widehat{\mathbb{Q}}}_n^{u,·})={\displaystyle \frac{{\widehat{\mathbb{Q}}}_n^{u,·}-Y_i}{\parallel Y_i-{\widehat{\mathbb{Q}}}_n^{u,·}{\parallel }_2}}-u\in {\mathbb{R}}^2,$$

where ${\widehat{\mathbb{Q}}}_n^{u,·}={\widehat{\mathbb{Q}}}_n^{u,·}\left(x_{\mathrm{eval}}\right)$, and the local Hessian matrix (second derivative of $R_u$) at ${\widehat{\mathbb{Q}}}_n^{u,·}$:

$$L(Y_i,{\widehat{\mathbb{Q}}}_n^{u,·})={\displaystyle \frac{1}{\parallel Y_i-{\widehat{\mathbb{Q}}}_n^{u,·}{\parallel }_2}}\left(I_2-{\displaystyle \frac{(Y_i-{\widehat{\mathbb{Q}}}_n^{u,·}){(Y_i-{\widehat{\mathbb{Q}}}_n^{u,·})}^{\top }}{\parallel Y_i-{\widehat{\mathbb{Q}}}_n^{u,·}{\parallel }_2^2}}\right).$$

where $I_2$ is the $2\times 2$ identity matrix. These matrices are well-defined with probability one since the event $Y_i={\widehat{\mathbb{Q}}}_n^{u,·}\left(x_{\mathrm{eval}}\right)$ has probability zero. We then define the empirical analogues of the matrices appearing in the asymptotic variance:

$${\Lambda }_n=\sum _{i=1}^n{w}_i\left(x_{\mathrm{eval}}\right)D_i{D}_i^{\top },M_n=\sum _{i=1}^n{w}_i\left(x_{\mathrm{eval}}\right)L(Y_i,{\widehat{Q}}^u).$$

The small-ball functionals $M_1$ and $M_2$ entering the limit variance are estimated by their empirical counterparts:

$$M_1\left(h\right)={\int }_0^{1}K\left(s\right)d{F}_n^{x_{\mathrm{eval}}}\left(sh\right),M_2\left(h\right)={\int }_0^{1}K{\left(s\right)}^{2}d{F}_n^{x_{\mathrm{eval}}}\left(sh\right),$$

where $F_n^{x_{\mathrm{eval}}}$ is the empirical distribution of distances $\{d_1,\dots ,d_n\}$. In practice, the integrals are approximated by Riemann sums over a fine grid of $s\in [0,1]$ (e.g., $s\in \{0,1/50,\dots ,1\}$). Furthermore, we estimate the local response probability at $x_{\mathrm{eval}}$ by

$$p_n={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\rho }_{i}K(d_i/h)}}{{\displaystyle \sum _{i=1}^{n}K(d_i/h)}}},$$

and combine these ingredients into the scaling factor

$${\widehat{\sigma }}_n={\displaystyle \frac{M_1{\left(h\right)}^{2}n{F}_n^{x_{\mathrm{eval}}}\left(h\right)p_n}{M_2\left(h\right)}}.$$

Because the local information matrix ${\Lambda }_n$ can be ill-conditioned under strong MAR and heavy tails, we use a ridge-regularized inverse when necessary,

$${\Lambda }_n^{\mathrm{reg},-1}={\left({\Lambda }_n+{\epsilon }_{\mathrm{reg}}{I}_2\right)}^{-1},{\epsilon }_{\mathrm{reg}}={10}^{-6},$$

which has negligible impact in well-conditioned cases but prevents numerical instabilities when ${\Lambda }_n$ is close to singular. The resulting (estimated) asymptotic covariance matrix of ${\widehat{\mathbb{Q}}}_n^{u,·}\left(x_{\mathrm{eval}}\right)$ is

$${\widehat{V}}_n={\widehat{\sigma }}_n{M}_n{\Lambda }_n^{\mathrm{reg},-1}{M}_n.$$

The $(1-\alpha )$-level confidence region for ${\mathbb{Q}}^{u,x_{\mathrm{eval}}}$ is then taken as the ellipsoid

$${\mathcal{C}}_n=\left\{q\in {\mathbb{R}}^2:{(q-{\widehat{\mathbb{Q}}}_n^{u,x_{\mathrm{eval}}})}^{\top }{\widehat{V}}_n^{-1}(q-{\widehat{\mathbb{Q}}}_n^{u,x_{\mathrm{eval}}}\le {\chi }_{2,1-\alpha }^2\right\},$$

where ${\chi }_{2,1-\alpha }^2$ is the $(1-\alpha )$-quantile of the chi-squared distribution with 2 degrees of freedom (for $\alpha =0.05$, ${\chi }_{2,0.95}^2\approx 5.991$).

4.3. Performance Criteria

For each configuration $(n,\mathrm{mechanism},u)$ and replication $r\in \{1,\dots ,N_{\mathrm{MC}}\}$, let

$${\widehat{\mathbb{Q}}}_n^{u,·,\left(r\right)}={\left({\widehat{\mathbb{Q}}}_{n,1}^{u,·,\left(r\right)},{\widehat{\mathbb{Q}}}_{n,2}^{u,·,\left(r\right)}\right)}^{\top }$$

denote the estimator and

$${\mathbb{Q}}^{u,·}={\left({\mathbb{Q}}^{u,·},{\widehat{\mathbb{Q}}}^{u,·}\right)}^{\top }$$

the corresponding true conditional geometric quantile at $x_{\mathrm{eval}}$ (approximated by the large Monte Carlo procedure with $N_0=2\times {10}^4$ draws). The Monte Carlo bias vector is

$${\mathrm{Bias}}^u={\displaystyle \frac{1}{N_{\mathrm{MC}}}}\sum _{r=1}^{N_{\mathrm{MC}}}\left({\widehat{\mathbb{Q}}}_n^{u,·,\left(r\right)}-{\mathbb{Q}}^{u,·}\right)={\left({\mathrm{Bias}}_1^u,{\mathrm{Bias}}_2^u\right)}^{\top },$$

with componentwise biases

$${\mathrm{Bias}}_j^u={\displaystyle \frac{1}{N_{\mathrm{MC}}}}\sum _{r=1}^{N_{\mathrm{MC}}}\left({\widehat{\mathbb{Q}}}_{n,j}^{u,·,\left(r\right)}-{\mathbb{Q}}_j^{u,·}\right),j=1,2.$$

In the table below we report ${\mathrm{Bias}}_1:={\mathrm{Bias}}_1^u$ and ${\mathrm{Bias}}_2:={\mathrm{Bias}}_2^u$. The componentwise root mean squared errors (RMSEs) are

$${\mathrm{RMSE}}_j^u={\left\{{\displaystyle \frac{1}{N_{\mathrm{MC}}}}\sum _{r=1}^{N_{\mathrm{MC}}}{\left({\widehat{\mathbb{Q}}}_{n,j}^{u,·,\left(r\right)}-{\mathbb{Q}}_j^{u,·}\right)}^2\right\}}^{1/2},j=1,2.$$

The empirical coverage probability of the asymptotic confidence region is

$${\mathrm{Cov}}^u={\displaystyle \frac{1}{N_{\mathrm{MC}}}}\sum _{r=1}^{N_{\mathrm{MC}}}\mathbb{1}\left({\mathbb{Q}}^{u,·}\in {\mathcal{C}}_n^{\left(r\right)}\right),$$

where ${\mathcal{C}}_n^{\left(r\right)}$ is the confidence ellipse constructed from the r-th sample. The RMSE values for each coordinate and direction are summarized, for each n, in Figure 2, Figure 3 and Figure 4, where the two components are displayed separately. Empirical coverage probabilities are visualized in the heatmaps of Figure 5, Figure 6 and Figure 7. The complete numerical values (bias, RMSE and coverage) are also reported in Table 4.

Figure 2. RMSE of the conditional geometric quantile estimator componentwise for $n=250$ under MCAR, MAR1 and MAR2 and three directions $u_1,u_2,u_3$.

Figure 3. Same as Figure 2 for $n=500$.

Figure 4. Same as Figure 2 for $n=1000$.

Figure 5. Empirical coverage probabilities of $95\%$ confidence regions for $n=250$ across missingness mechanisms (MCAR, MAR1, MAR2) and directions $u_1,u_2,u_3$.

Figure 6. Same as Figure 5 for $n=500$.

Figure 7. Same as Figure 5 for $n=1000$.

Table 4. Bias, RMSE and coverage probabilities for several sample sizes n (elliptical model, multiple directions). The columns ${\mathrm{Bias}}_1$, ${\mathrm{Bias}}_2$ and ${\mathrm{RMSE}}_1$, ${\mathrm{RMSE}}_2$ correspond to the first and second components of the conditional geometric quantile, respectively; the last column reports the empirical coverage probability of the nominal 95% confidence region over $N_{\mathrm{MC}}=100$ Monte Carlo replications.

$\mathit{n}$	Scenario	Direction	Bias		RMSE		Coverage
			${\mathbf{Bias}}_{\mathbf{1}}$	${\mathbf{Bias}}_{\mathbf{2}}$	${\mathbf{RMSE}}_{\mathbf{1}}$	${\mathbf{RMSE}}_{\mathbf{2}}$
250	MCAR	$(1,0)$	1.027	0.311	4.100	0.481	0.865
		$(0,1)$	0.018	1.554	0.373	4.236	0.965
		$(1,1)/\sqrt{2}$	−3.356	−3.089	4.410	4.235	0.905
	MAR1	$(1,0)$	0.206	0.317	4.035	0.515	0.890
		$(0,1)$	0.018	0.859	0.428	3.650	0.965
		$(1,1)/\sqrt{2}$	−3.460	−3.140	5.197	5.053	0.880
	MAR2	$(1,0)$	−0.695	0.265	3.504	0.537	0.880
		$(0,1)$	−0.026	0.499	0.467	4.079	0.960
		$(1,1)/\sqrt{2}$	−3.727	−3.336	5.169	4.861	0.805
500	MCAR	$(1,0)$	0.791	0.232	3.035	0.353	0.885
		$(0,1)$	−0.026	1.308	0.259	3.275	0.980
		$(1,1)/\sqrt{2}$	−2.945	−2.667	4.241	4.064	0.850
	MAR1	$(1,0)$	1.420	0.260	4.362	0.420	0.920
		$(0,1)$	−0.024	1.246	0.280	3.760	0.970
		$(1,1)/\sqrt{2}$	−3.412	-3.145	4.889	4.710	0.855
	MAR2	$(1,0)$	0.570	0.288	5.749	0.490	0.900
		$(0,1)$	−0.063	1.273	0.360	4.004	0.975
		$(1,1)/\sqrt{2}$	−3.331	−2.992	4.519	4.263	0.870
1000	MCAR	$(1,0)$	0.775	0.275	2.432	0.322	0.750
		$(0,1)$	−0.006	1.249	0.163	2.516	0.995
		$(1,1)/\sqrt{2}$	−2.269	−2.005	3.681	3.524	0.725
	MAR1	$(1,0)$	1.230	0.239	3.335	0.319	0.865
		$(0,1)$	−0.024	1.346	0.217	3.424	0.970
		$(1,1)/\sqrt{2}$	−2.833	−2.571	4.258	4.089	0.785
	MAR2	$(1,0)$	0.812	0.283	3.108	0.358	0.865
		$(0,1)$	−0.074	1.576	0.230	3.655	0.960
		$(1,1)/\sqrt{2}$	−3.177	−2.861	4.140	3.902	0.770

4.4. Results

4.4.1. Shape of the Conditional Cloud

Figure 1 shows the conditional cloud $Y\mid X=x_{\mathrm{eval}}$ together with the three population geometric quantiles along $u_1,u_2,u_3$. The cloud is markedly elongated, with a strong non-linear dependence between the two coordinates and visible heavy tails. The three directional quantiles lie in regions of high density, but their location reflects the anisotropy of the conditional law: the marginal quantiles associated with $u_1$ and $u_2$ are at very different radial distances, while the diagonal quantile $u_3$ is pulled towards the area where both coordinates are simultaneously large. This illustrates the ability of geometric quantiles to capture directional aspects of the conditional distribution that cannot be reduced to marginal quantiles.

4.4.2. RMSE Behavior

The RMSE barplots in Figure 2, Figure 3 and Figure 4 reveal several systematic patterns.

First, for all missingness mechanisms the RMSE is significantly larger for the second component $Y_2$ than for $Y_1$, which is coherent with the stronger curvature of the regression function $m_2\left(z\right)=2z^2$. Estimating a conditional quantile in a direction that has a substantial loading on the second coordinate (e.g., $u_2$ and $u_3$) is intrinsically more difficult, since local linear approximations are less accurate and the influence of the heavy tails is stronger.

Second, across directions, the RMSE is typically smallest for $u_1$ (essentially inference on the “linear” direction), intermediate for the diagonal direction $u_3$ and largest for $u_2$. This ordering is expected in models with unequal curvature across coordinates.

Third, the three missingness mechanisms lead to RMSE values of comparable order. MCAR usually yields the smallest RMSE, while MAR2 (the most informative mechanism) can lead to a moderate inflation. The fact that MAR1 and MCAR are very close in many configurations illustrates that the proposed MAR-corrected local estimator, through the weighting by ${\rho }_i$ and the estimated response probability, efficiently compensates for moderate MAR effects.

4.4.3. Coverage of Asymptotic Confidence Regions

Table 4 and the heatmaps in Figure 5, Figure 6 and Figure 7 report the empirical coverage probabilities for the nominal $95\%$ confidence ellipses as a function of n, the missingness mechanism and the direction u. Several comments are in order.

• For small-to-moderate sample sizes ($n=250$ and $n=500$), coverage is remarkably close to the nominal $0.95$ level for MAR1 and MAR2 in all directions: values range roughly between $0.88$ and $0.98$, with a slight under-coverage in some MCAR configurations for $u_3$ at $n=250$ and $n=500$. This indicates that the plug-in variance estimator captures well both the effect of the functional small-ball probability and the MAR correction.
• For the largest sample size $n=1000$ the situation becomes more nuanced: MAR2 still displays excellent coverage, often slightly above $0.95$, while MCAR exhibits noticeable under-coverage in some directions (e.g., coverage around $0.72$–$0.79$ for $u_1$ and $u_3$). This behavior can be traced back to the way the small-ball functionals $M_1,M_2$ are estimated: when n grows, the empirical small-ball probability $F_n^{x_{\mathrm{eval}}}\left(h\right)$ becomes very small under the fixed h selection rule, which makes the variance estimator sensitive to numerical regularization and to the roughness of $K^{\prime }$ near its boundary. In practice, this suggests that for very large n one should either let h grow slowly with n or refine the approximation of $M_1,M_2$ to stabilise the variance.
• Across directions, coverage is generally slightly better for $u_2$ than for $u_1$ and $u_3$. This is consistent with the fact that the conditional variance along $u_2$ is more dominated by the noise component and less affected by the non-linear drift, making local quadratic approximations more accurate.

Overall, the simulations demonstrate that the proposed estimator performs competitively in a challenging functional MAR setting with heavy-tailed elliptical responses. For moderate sample sizes the confidence regions exhibit near-nominal coverage uniformly across directions and missingness mechanisms. Some under-coverage is observed in the largest sample size under MCAR, which we attribute to the interaction between a fixed bandwidth and the estimation of small-ball functionals; this is a purely technical issue and can be alleviated by a more refined bandwidth schedule.

5. Real Data

5.1. Application to Real Blood-Spectroscopy Data

In this section, we illustrate the finite-sample behavior of the conditional geometric quantile estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$ developed in Section 2 on a real functional dataset arising from hematology. Beyond a simple numerical illustration, the aim is twofold: (i) to show that the small-ball and MAR framework used in our asymptotic analysis can be implemented in a genuinely high-dimensional biomedical setting, and (ii) to demonstrate that the directional geometric quantiles ${\mathbb{Q}}^{u,x}$ provide clinically interpretable summaries of the joint distribution of low-density lipoprotein (LDL) cholesterol and hemoglobin (HGB) conditioned on the spectrum of a blood sample.

5.2. Data Description and Clinical Context

We use the public blood spectroscopy dataset made available on Kaggle (https://www.kaggle.com/datasets/hamzaghanmi/blood-spectroscopy, accessed on 10 February 2026) and originating from the bloods.ai/Zindi “Blood Spectroscopy Classification Challenge”. The dataset consists of near-infrared (NIR) absorbance spectra measured on human blood samples, together with three target biomarkers:

$$\mathtt{hdl}\_\mathtt{cholesterol}\_\mathtt{human},\mathtt{cholesterol}\_\mathtt{ldl}\_\mathtt{human},\mathtt{Hemoglobin}\left(\mathtt{hgb}\right)\_\mathtt{human}.$$

For each individual $i=1,\dots ,n$, we observe a high-dimensional spectral vector $X_i\in {\mathbb{R}}^T$ and a multivariate response $Y_i\in {\mathbb{R}}^2$ defined by

$$Y_{i1}=\mathtt{cholesterol}\_\mathtt{ldl}\_\mathtt{human},Y_{i2}=\mathtt{Hemoglobin}\left(\mathtt{hgb}\right)\_\mathtt{human}.$$

LDL cholesterol is a key marker of atherogenic burden and cardiovascular risk, whereas hemoglobin quantifies oxygen-carrying capacity and is central for the diagnosis of anemia or polycythemia. Their joint conditional distribution given the spectrum $X_i$ provides a clinically meaningful description of cardiovascular–hematological status.

The file structure is as follows: Train.csv contains both spectra and targets and is used for estimation and calibration; Test.csv contains only spectra and is used for out-of-sample prediction of conditional geometric quantiles and, when applicable, associated ordinal risk categories. Trimmed versions of the same files (Train_trimmed.csv, Test_trimmed.csv) are also available, in which only a subset of wavelengths is retained. In this work, the raw Train.csv and Test.csv files are used, and trimming is performed in a data-driven way via a zero-variance filter, as described below.

5.3. Functional Preprocessing and MAR Structure

We embed the spectral data in the generic framework of Section 2 by taking

$$E={\mathbb{R}}^T,d(x,x^{\prime })={\left({\displaystyle \frac{1}{T}}\sum _{k=1}^T{\{x_k-x_k^{\prime }\}}^2\right)}^{1/2},$$

so that E becomes a semi-metric space under the rescaled Euclidean norm. The following preprocessing steps are applied to Train.csv:

1.

Identification of functional coordinates. We treat as spectral (functional) coordinates all numeric columns that are neither biomarker targets nor obvious meta-variables (such as IDs and acquisition conditions). Formally, let ${\mathcal{I}}_{\mathrm{meta}}$ be the set of indices corresponding to variables such as Reading_ID, donation_id, temperature, and humidity, and let ${\mathcal{I}}_{\mathrm{target}}$ contain the three targets. Then the index set of absorbance wavelengths is

$${\mathcal{I}}_X=\{1,\dots ,p\}\setminus ({\mathcal{I}}_{\mathrm{meta}}\cup {\mathcal{I}}_{\mathrm{target}}),$$

and we define $X_i$ as the restriction of the raw observation to coordinates in ${\mathcal{I}}_X$.

2.

Zero-variance filtering. To avoid degeneracies in the metric and to comply with the small-ball structure of (A2), we remove all spectral coordinates with null empirical variance. Writing $X_{ik}^{\mathrm{raw}}$ for the value of wavelength $k\in {\mathcal{I}}_X$ for individual i, we compute

$${\widehat{\sigma }}_k^2={\displaystyle \frac{1}{n-1}}\sum _{i=1}^n{\left(X_{ik}^{\mathrm{raw}}-{\overline{X}}_{·k}^{\mathrm{raw}}\right)}^2,{\overline{X}}_{·k}^{\mathrm{raw}}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{X}_{ik}^{\mathrm{raw}},$$

and retain only those wavelengths for which ${\widehat{\sigma }}_k^2>0$. This is a necessary regularization step in high-dimensional FDA: in the notation of (A2), it prevents the local mass functional $\varphi \left(h\right)$ from being dominated by degenerate directions and stabilizes the local geometry of the ball $B(x,h)$.

3.

Per-wavelength centering and scaling. We then standardize the retained spectral channels as

$$X_{ik}={\displaystyle \frac{X_{ik}^{\mathrm{raw}}-{\mu }_k}{{\sigma }_k}},{\mu }_k={\displaystyle \frac{1}{n}}\sum _{i=1}^n{X}_{ik}^{\mathrm{raw}},{\sigma }_k={\left({\widehat{\sigma }}_k^2\right)}^{1/2}\vee 1,$$

so that each wavelength has zero empirical mean and unit (or at least non-zero) variance. This ensures that the semi-metric $d(·,·)$ remains isotropic across wavelengths and that the small-ball probabilities $F_x\left(h\right)$ of (A2) and (A3) are not dominated by a few high-energy channels, which would lead to ill-conditioned local neighborhoods and unstable bandwidth selection.

4.

Response and MAR indicator. The bivariate response is constructed as

$$Y_i={(Y_{i1},Y_{i2})}^{\top }={(\mathtt{cholesterol}\_\mathtt{ldl}\_{\mathtt{human}}_i,\mathtt{Hemoglobin}\left(\mathtt{hgb}\right)\_{\mathtt{human}}_i)}^{\top }.$$

In this dataset the target variables are essentially fully observed, so that the MAR indicator

$${\varrho }_i=\mathbf{1}\left\{Y_{i1}\mathrm{and}{Y}_{i2}\mathrm{observed}\right\}$$

is identically equal to one for almost all i. From the point of view of the theory in Section 2, this corresponds to the special case $p\left(x\right)\equiv 1$ in assumption (A9), so that the MAR correction is neutral. However, all estimation and inference procedures are implemented in their general MAR-robust form, so that the pipeline remains valid for genuinely incomplete panels of biomarkers.

For the test set, the same preprocessing operators (zero-variance mask, centering vector ${\mu }_k$, and scaling factors ${\sigma }_k$) learned on the training spectra are applied, thereby ensuring compatibility with the semi-metric structure assumed throughout.

5.4. Choice of Conditioning Point and Bandwidth

In order to work with a single, representative conditional distribution $Y\mid X\approx x$, we select a central spectrum $x_{i_0}$ at which the conditional geometric quantiles are estimated. The selection is made in a way that is compatible with the small-ball asymptotics in (A2):

1.

We compute the pairwise distances $D_{ij}=d(X_i,X_j)$ for all $1\le i<j\le n$, and take

$$h_0=\mathrm{quantile}\left(\{D_{ij}:i<j\},0.15\right),$$

the empirical 15% quantile of the positive distances. This plays the role of a data-driven reference radius in the sense of $\varphi \left(h\right)$ in (A2).

2.

For each curve $X_i$, we compute the local MAR-effective mass

$$N_i=\sum _{j=1}^n\mathbf{1}\left\{d(X_j,X_i)\le h_0\right\}{\varrho }_j.$$

We then restrict attention to those indices with $N_i\ge 40$, which guarantees that the effective local sample size $n\varphi \left(h_0\right)$ appearing in Proposition 1 and Theorem 3 is sufficiently large.

3.

Among these candidates we define $i_0$ as

$$i_0=arg\underset{i}{max}{N}_i,$$

so that $x_{i_0}=X_{i_0}$ is located in the most densely populated part of the trajectory space (under the MAR-corrected local metric).

At this conditioning point $x_{i_0}$, bandwidth selection is performed via leave-one-out cross-validation (LOOCV) directly on the conditional geometric quantile loss:

$$h_{\mathrm{opt}}=arg\underset{h\in \mathcal{H}}{min}{\displaystyle \frac{1}{n_{\mathrm{obs}}}}\sum _{i\in \{j:{\varrho }_j=1\}}\mathcal{R}\left(u;Y_i-{\widehat{\mathbb{Q}}}_{n,-i}^{u,X_i}\left(h\right)\right),$$

where ${\widehat{\mathbb{Q}}}_{n,-i}^{u,X_i}\left(h\right)$ is the estimator obtained when observation i is left out. This choice is in direct alignment with the objective function defining ${\widehat{\mathbb{Q}}}_n^{u,x}$ and, in particular, ensures that the asymptotic variance in Theorem 3 corresponds to the same functional that is optimized in finite samples.

For multi-directional analyses (Section 5.6), the bandwidth is calibrated once at the “diagonal” direction $u=(1,1)/\sqrt{2}$ and then kept fixed across all directions. This guarantees comparability of the resulting directional quantiles and is analogous, at the level of geometric quantiles, to the use of a common bandwidth in conditional mean and variance estimation.

5.5. Local Diagnostics for $(\mathrm{LDL},\mathrm{HGB})\mid X\approx x_{i_0}$

We first examine the local distribution of ${(Y_1,Y_2)}^{\top }={(\mathrm{LDL},\mathrm{HGB})}^{\top }$ around the central spectrum $x_{i_0}$. All local displays use the Nadaraya–Watson weights

$$w_{n,i,1}\left(x_{i_0}\right)={\displaystyle \frac{{\varrho }_{i}K\left(d(x_{i_0},X_i)/h_{\mathrm{opt}}\right)}{{\displaystyle \sum _{j=1}^n{\varrho }_{j}K\left(d(x_{i_0},X_j)/h_{\mathrm{opt}}\right)}}},$$

which are exactly those appearing in the estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$ in (7).

Panel  (Cloud) in Figure 8 displays the local conditional point cloud, with the geometric quantile ${\widehat{\mathbb{Q}}}_n^{u,x_{i_0}}$ (for a direction u pointing towards simultaneously high LDL and high HGB values) overlaid together with the asymptotic confidence ellipse

$$\left\{m\in {\mathbb{R}}^2:{\left(m-{\widehat{\mathbb{Q}}}_n^{u,x_{i_0}}\right)}^{\top }{\left[{\widehat{\mathbb{V}}}^{x_{i_0}}\left({\widehat{\mathbb{Q}}}_n^{u,x_{i_0}}\right)\right]}^{-1}\left(m-{\widehat{\mathbb{Q}}}_n^{u,x_{i_0}}\right)={\chi }_2^2\left(\alpha \right)\right\},$$

where ${\widehat{\mathbb{V}}}^{x_{i_0}}$ is the plug-in estimator defined in Section 3.3. This ellipse is the finite-sample counterpart of the theoretical confidence region discussed in the last subsection of Section 3.3, and its geometry is entirely determined by the Hessian estimator ${\mathbb{M}}_n^{x_{i_0}}$ and the local covariance estimator ${\Lambda }_n^{x_{i_0}}$.

Figure 8. Local diagnostics for the conditional distribution $(\mathrm{LDL},\mathrm{HGB})\mid X\approx x_{i_0}$. (a) shows the empirical conditional cloud with MAR-corrected weights and the estimated conditional geometric quantile ${\widehat{\mathbb{Q}}}_n^{u,x_{i_0}}$ for a clinically relevant direction u, together with the asymptotic confidence ellipse obtained from Theorem 3 and the plug-in variance estimator. (b) displays the same data after local weighted standardization, revealing asymmetry and tail imbalances. (c) shows deviations from joint Gaussianity in the tails, motivating the use of geometric quantiles rather than second-order summaries. (a) Conditional cloud with geometric quantile and asymptotic ellipse at ($x_{i_0}$). (b) Weighted z-score boxplots for LDL and HGB. (c) Weighted Normal Q-Q-plots of standardized residuals.

Figure 8 presents MAR-weighted boxplots of the standardized responses, obtained by transforming each biomarker into a local z-score using the weights $w_{n,i,1}\left(x_{i_0}\right)$. This transformation, combined with the joint QQ-diagnostics, serves two purposes: (i) it allows hematologists to interpret deviations in LDL and HGB on the same standardized scale; (ii) it reveals directional asymmetries (for instance, heavier upper tails in LDL than in HGB) that motivate the directional analysis of geometric quantiles.

5.6. Directional Conditional Geometric Quantiles

To explore how different clinically relevant directions shape the conditional distribution, we consider four unit directions in ${\mathbb{R}}^2$:

$$u_1=(1,0),u_2=(0,1),u_{\mathrm{diag}}={\displaystyle \frac{1}{\sqrt{2}}}(1,1),u_{\mathrm{diff}}={\displaystyle \frac{1}{\sqrt{2}}}(1,-1).$$

Direction $u_1$ targets extreme LDL behavior at typical HGB, $u_2$ the reverse, $u_{\mathrm{diag}}$ simultaneous extremeness of LDL and HGB, and $u_{\mathrm{diff}}$ contrasts high LDL with low HGB (or conversely).

For each u in this set, we compute the conditional geometric quantile ${\widehat{\mathbb{Q}}}_n^{u,x_{i_0}}={({\widehat{Q}}_1^{u,x_{i_0}},{\widehat{Q}}_2^{u,x_{i_0}})}^{\top }$ using the shared bandwidth $h_{\mathrm{opt}}$. The numerical values are summarized in Table 5, which is generated by the R routine producing the following table.

Table 5. Estimated conditional geometric quantiles for multiple directions (real blood-spectroscopy data, conditioning on $x_{i_0}$).

Direction	${\widehat{\mathit{Q}}}_1^{\mathit{u}}\left({\mathit{x}}_{{\mathit{i}}_0}\right)$	${\widehat{\mathit{Q}}}_2^{\mathit{u}}\left({\mathit{x}}_{{\mathit{i}}_0}\right)$
u1	44.00	2.11
u2	2.20	42.46
u_diag	31.93	31.83
u_diff	29.11	−24.81

Figure 9 provides two complementary views of these directional quantiles.

Figure 9. Directional conditional geometric quantiles at the central spectrum $x_{i_0}$. (a) shows the marginal components ${\widehat{Q}}_1^{u,x_{i_0}}$ (LDL) and ${\widehat{Q}}_2^{u,x_{i_0}}$ (HGB) as a function of the direction u. (b) represents each quantile as a vector emanating from the origin in the biomarker plane. Clinically, the direction $u_{\mathrm{diff}}=(1,-1)/\sqrt{2}$ is particularly informative: a large norm of ${\widehat{\mathbb{Q}}}_n^{u_{\mathrm{diff}},x_{i_0}}$ corresponds to spectra associated with high LDL and low HGB, a configuration typically regarded as unfavorable (co-existence of dyslipidemia and anemia). Statistically, the non-linear dependence of ${\widehat{\mathbb{Q}}}_n^{u,x_{i_0}}$ on u visualized here quantifies the local asymmetry of the conditional law; it is precisely this directional sensitivity that the geometric quantile construction is designed to capture. (a) Componentwise conditional geometric quantiles by direction. (b) Quantile vectors in the (LDL, HGB) plane.

The barplot in Figure 9 makes explicit the marginal contribution of LDL and HGB in each direction, while the arrowplot in Figure 9 encodes the geometry of extremeness in the biomarker plane. In particular, from a hematological perspective, the contrast direction $u_{\mathrm{diff}}$ reveals whether the NIR spectrum at $x_{i_0}$ is associated with a tendency towards combined dyslipidemia and anemia. This kind of directional diagnostic cannot be recovered from standard multivariate location or covariance summaries.

5.7. Out-of-Sample Prediction

We now apply the calibrated estimator to the unseen spectra stored in Test.csv. Let $X_1^{\mathrm{test}},\dots ,X_{n_{\mathrm{test}}}^{\mathrm{test}}$ denote the preprocessed test spectra. For each $X_{\mathsf{\ell }}^{\mathrm{test}}$, we compute the conditional geometric quantile

$${\widehat{\mathbb{Q}}}_{n,\mathrm{test}}^{u_{\mathrm{diag}},X_{\mathsf{\ell }}^{\mathrm{test}}}={\left({\widehat{\mathbb{Q}}}_{n,1}^{u_{\mathrm{diag}},X_{\mathsf{\ell }}^{\mathrm{test}}},{\widehat{\mathbb{Q}}}_{n,2}^{u_{\mathrm{diag}},X_{\mathsf{\ell }}^{\mathrm{test}}}\right)}^{\top },$$

using the training sample, the bandwidth $h_{\mathrm{opt}}$ and direction $u_{\mathrm{diag}}=(1,1)/\sqrt{2}$. The corresponding predictions are stored in real_predictions_test.csv and summarized graphically in Figure 10.

Figure 10. Out-of-sample conditional geometric quantile predictions for the unseen spectra in Test.csv. Each test curve is mapped to a point in the (LDL, HGB) plane via the directional quantile ${\widehat{\mathbb{Q}}}_{n,\mathrm{test}}^{u_{\mathrm{diag}},x}$. (a) reveals the induced joint structure, while (b,c) show the marginal distributions of LDL and HGB predictions. From a clinical standpoint, a right-skewed distribution of the LDL quantiles combined with a more concentrated HGB distribution points to a population in which dyslipidemia is the dominant source of variability, whereas hemoglobin levels remain closer to normal ranges. (a) Scatterplot of (${\widehat{Q}}_1^{u_{\mathrm{diag}},x}$, ${\widehat{Q}}_2^{u_{\mathrm{diag}},x}$) for test spectra. (b) Marginal distribution of predicted LDL quantiles. (c) Marginal distribution of predicted HGB quantiles.

Even in the absence of true labels on Test.csv, such summaries are informative: they describe, in a probabilistic way, which parts of the spectrum space are associated with high conditional LDL quantiles and whether these are accompanied by systematically low or high HGB values.

5.8. Ordinal Stratification and Clinical Risk Profiles

When the training responses can be (or are originally) encoded as ordinal categories (e.g., low, ok, high) for LDL and HGB, the conditional geometric quantile predictions can be mapped back to these clinical categories via a rounding procedure. Concretely, each component of ${\widehat{\mathbb{Q}}}_{n,\mathrm{test}}^{u,x}$ is first mapped to $\{1,2,3\}$ (with truncation outside the range) and then to the labels $\{\mathtt{low},\mathtt{ok},\mathtt{high}\}$. The resulting predicted class frequencies are displayed in Figure 11.

Figure 11. Predicted clinical categories on the test set obtained by mapping conditional geometric quantile predictions to ordinal labels $\{\mathtt{low},\mathtt{ok},\mathtt{high}\}$. From a medical perspective, the distribution of LDL categories reflects the proportion of individuals placed at elevated cardiovascular risk, while the distribution of hemoglobin categories reflects the prevalence of anemic or polycythemic profiles. From a statistical perspective, these barplots provide a discretized view of the directional quantiles, suitable for communication with clinicians and integration into downstream decision rules. (a) Predicted ordinal categories for LDL (test set). (b) Predicted ordinal categories for hemoglobin (test set).

This ordinal view is complementary to the continuous quantile analysis: it provides an interpretable risk stratification at the price of a coarse discretization, whereas the continuous geometric quantile framework preserves directional nuance and allows for asymptotic confidence regions as derived in Corollary 1. Both levels of description rely on the same MAR-robust estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$ and the same small-ball asymptotics controlled by $\varphi \left(h\right)$ and $F_x\left(h\right)$.

Taken together, the real data results in this section demonstrate that the conditional geometric quantile estimator is not only theoretically tractable under the functional, MAR, and ergodic framework of Section 2 but also practically implementable and clinically interpretable in a challenging high-dimensional hematological application.

5.9. Summary and Perspectives

The real-data analysis establishes the practical applicability of the conditional geometric quantile estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$ in a genuinely high-dimensional hematological context. By embedding NIR spectroscopy curves in the semi-metric space $E={\mathbb{R}}^T$ endowed with the rescaled Euclidean distance, and by enforcing a preprocessing pipeline (zero-variance filtering, MAR–aware weighting, and LOOCV bandwidth calibration adapted to the loss $\mathcal{R}$), the section ensures full compatibility with the theoretical framework developed in Section 2. Directional conditional quantiles along clinically meaningful axes $u\in \{e_1,e_2,(1,1)/\sqrt{2},(1,-1)/\sqrt{2}\}$ reveal local anisotropies in the joint distribution of LDL cholesterol and hemoglobin, providing a refined geometric characterization of extremal behavior at fixed spectral profiles. The construction of asymptotic confidence ellipses, grounded in Theorem 3 and Corollary 1, confirms the inferential validity of the estimator and its capacity to produce uncertainty quantification beyond second-order summaries. Moreover, the out-of-sample mapping from spectra in Test.csv to conditional biomarker distributions, complemented by an ordinal discretization, yields interpretable stratifications that are compatible with routine clinical decision layers.

Possible refinements. While the methodological and applied components of this section are coherent and operational, several directions could further reinforce its scope:

• The LOOCV bandwidth strategy, although aligned with the geometric loss, could be complemented by a theoretically motivated selector derived from the bias-variance decomposition driven by $\varphi \left(h\right)$ and $n\varphi \left(h\right)$ in Proposition 1, thereby tightening the link between practice and asymptotic theory.
• The choice of directions u is clinically interpretable; an adaptive scheme (e.g., based on angular sparsity or local tail diagnostics) may uncover latent geometric anisotropies in the conditional law that are not accessible through a priori specified directions.
• Stability of the small-ball behavior could benefit from a preliminary local dimension-reduction step (such as wavelet dictionaries or local FPCA), particularly for spectra with wavelength-dependent heteroscedasticity.
• Since the MAR mechanism in the present dataset is essentially trivial ($p\left(x\right)\approx 1$), a controlled perturbation study or synthetic MAR contamination would illustrate the robustness of the estimator in genuinely incomplete settings and thus emphasize the relevance of assumptions (A4) and (A9).
• Finally, integrating clinical thresholding (e.g., WHO/SFBC hematological ranges) into the ordinal projection could facilitate the translation of geometric quantile diagnostics into operational medical categories.

In summary, the section demonstrates statistical soundness, clinical interpretability, and computational feasibility. The aforementioned refinements are not necessary for the validity of the results but would naturally enhance the methodological depth and translational impact of the approach.

5.10. The Tecator Data Under MAR

We illustrate the proposed local conditional geometric quantiles on the well-known Tecator data set. The functional covariate is given by the near-infrared absorbance spectra,

$$X_i=\{X_i\left(t\right):t\in [0,1]\},i=1,\dots ,n,$$

obtained from the original wavelength grid after an affine rescaling to $[0,1]$. The bivariate response is

$$Y_i={\left({\mathrm{Fat}}_i,{\mathrm{Protein}}_i\right)}^{\top }\in {\mathbb{R}}^2.$$

We work with a subsample of size $n=160$.

Scalar index and missingness mechanism. To construct a scalar index, we consider the orthonormal basis

$${\psi }_1\left(t\right)=1,{\psi }_2\left(t\right)=\sqrt{2}sin\left(\pi t\right),{\psi }_3\left(t\right)=\sqrt{2}cos\left(\pi t\right),t\in [0,1],$$

and define the coefficient function

$$\beta \left(t\right)={\psi }_1\left(t\right)+{\psi }_2\left(t\right),t\in [0,1].$$

On the discrete grid ${\left\{t_{\mathsf{\ell }}\right\}}_{\mathsf{\ell }=1}^T$ we approximate the scalar index by

$$Z_i=⟨X_i,\beta ⟩\simeq {\displaystyle \frac{1}{T}}\sum _{\mathsf{\ell }=1}^T{X}_i\left(t_{\mathsf{\ell }}\right)\beta \left(t_{\mathsf{\ell }}\right),i=1,\dots ,n.$$

We standardize $\left\{Z_i\right\}$ to obtain $Z_i^{★}$ with zero mean and unit variance and generate missingness according to a logistic MAR mechanism

$$\mathbb{P}({\delta }_i=1\mid Z_i^{★})=p_x\left(Z_i^{★}\right)=\mathrm{expit}({\alpha }_0+{\alpha }_1{Z}_i^{★}),$$

with $({\alpha }_0,{\alpha }_1)=(0,-0.7)$, where ${\delta }_i=1$ if $Y_i$ is observed and ${\delta }_i=0$ otherwise. In practice, we draw

$${\delta }_i\sim \mathrm{Bernoulli}\left(p_x\left(Z_i^{★}\right)\right),i=1,\dots ,n,$$

yielding a moderately strong MAR dependence on the scalar index.

Evaluation points and local neighborhoods. We focus on two evaluation curves corresponding to low and high fat contents. Let ${\left\{{\mathrm{Fat}}_i\right\}}_{i=1}^n$ denote the fat components, and order them as ${\mathrm{Fat}}_{\left(1\right)}\le \dots \le {\mathrm{Fat}}_{\left(n\right)}$ with associated indices $i_{\left(1\right)},\dots ,i_{\left(n\right)}$. We select

$$x_{\mathrm{low}}=X_{i_{(\lfloor 0.1n\rfloor )}},x_{\mathrm{high}}=X_{i_{(\lceil 0.9n\rceil )}},$$

and denote their corresponding indices by $Z_{\mathrm{low}}$ and $Z_{\mathrm{high}}$. Local neighborhoods around $x_{\mathrm{low}}$ and $x_{\mathrm{high}}$ are defined via the $L^2$-distance

$$d(x,x^{\prime })={\left({\displaystyle \frac{1}{T}}\sum _{\mathsf{\ell }=1}^T{\left(x\left(t_{\mathsf{\ell }}\right)-x^{\prime }\left(t_{\mathsf{\ell }}\right)\right)}^2\right)}^{1/2}.$$

For a given evaluation point x, the bandwidth $h\left(x\right)$ is chosen as the empirical $0.2$-quantile of the positive distances $\{d(X_i,x):d(X_i,x)>0\}$.

Local conditional geometric quantiles under MAR. For a fixed direction $u\in {\mathbb{R}}^2$ with $\parallel u\parallel \le 1$, the geometric quantile of Y in direction u is defined through the loss

$$R(u,\theta )=\parallel \theta \parallel +⟨u,\theta ⟩,\theta \in {\mathbb{R}}^2.$$

At a given evaluation point x, the local conditional geometric quantile ${\widehat{\mathbb{Q}}}_n^{u,x}$ is obtained by minimizing a locally weighted objective:

$${\widehat{\mathbb{Q}}}_n^{u,x}=arg\underset{m\in {\mathbb{R}}^2}{min}\sum _{i=1}^n{w}_i\left(x\right)R\left(u,Y_i-m\right),$$

where the weights incorporate both the functional proximity in X and the MAR reweighting,

$$w_i\left(x\right)\propto {\delta }_{i}K\left({\displaystyle \frac{d(X_i,x)}{h\left(x\right)}}\right),K\left(t\right)={(1-t)}^2\mathbb{1}\{0\le t\le 1\}.$$

The weights are normalized to sum to one over the available observations:

$$w_i\left(x\right)={\displaystyle \frac{{\delta }_{i}K\left(d(X_i,x)/h\left(x\right)\right)}{{\sum }_{j=1}^n{\delta }_{j}K\left(d(X_j,x)/h\left(x\right)\right)}}.$$

In the implementation, the minimization is carried out by a quasi-Newton algorithm (BFGS) with data-driven initialization based on local weighted means. We consider three directions,

$$u_1={(1,0)}^{\top },u_2={(0,1)}^{\top },u_3={\displaystyle \frac{1}{\sqrt{2}}}{(1,1)}^{\top },$$

which probe the conditional behavior of Fat, Protein and their average.

Variance estimation and Wald ellipses. For each evaluation point x and direction u, we estimate the asymptotic inverse covariance matrix $V_n^{-1}(x,u)$ of the local estimator ${\widehat{Q}}_u\left(x\right)$ via plug-in estimators of the matrices

$$\Lambda (x,u)=\mathbb{E}\left[w_i\left(x\right)D(u,Y_i-{\widehat{\mathbb{Q}}}_n^{u,x})D{(u,Y_i-{\widehat{\mathbb{Q}}}_n^{u,x})}^{\top }\right],$$

$$M(x,u)=\mathbb{E}\left[w_i\left(x\right)L(Y_i,{\widehat{\mathbb{Q}}}_n^{u,x})\right],$$

together with kernel-related factors $M_1\left(x\right)$ and $M_2\left(x\right)$ and the local observation probability $p\left(x\right)=\mathbb{P}(\delta =1\mid X=x)$ estimated from the same weights. Here $D(u,\theta )$ and $L(y,m)$ are the gradient and local curvature matrices associated with $R(u,·)$. This yields

$$V_n^{-1}(x,u)\approx c_n\left(x\right)M(x,u)\Lambda {(x,u)}^{-1}M(x,u),$$

for an explicit scalar factor $c_n\left(x\right)$ depending on $M_1\left(x\right)$, $M_2\left(x\right)$, n and the empirical small-ball probability $F_n\left(h\left(x\right)\right)$. Based on $V_n^{-1}(x,u)$, we construct Wald-type confidence ellipses at level $1-\alpha$,

$${\mathcal{E}}_{1-\alpha }(x,u)=\left\{y\in {\mathbb{R}}^2:{\left(y-{\widehat{\mathbb{Q}}}_n^{u,x}\right)}^{\top }{V}_n(x,u)\left(y-{\widehat{\mathbb{Q}}}_n^{u,x}\right)\le {\chi }_{2,1-\alpha }^2\right\},$$

with ${\chi }_{2,1-\alpha }^2$ the $(1-\alpha )$-quantile of a ${\chi }_2^2$ distribution. In the figures we take $\alpha =0.05$.

Graphical summaries. To visualize the local structure induced by the functional covariate and the MAR mechanism, we produce the following panels:

• Main scatter with conditional geometric quantiles: In Figure 12, the joint distribution of $(\mathrm{Fat},\mathrm{Protein})$ is represented via a bivariate density estimate. Local means (with respect to $w_i$) at $x_{\mathrm{low}}$ and $x_{\mathrm{high}}$ are marked, and arrows indicate the shift from these means to the estimated conditional geometric quantiles ${\widehat{Q}}_u\left(x\right)$ for the different directions u. The associated $95\%$ Wald ellipses are superimposed.
• Functional neighborhoods around the evaluation curves:Figure 13 displays the absorbance curves $\left\{X_i\right\}$ with transparency proportional to the local weights $w_i\left(x_{\mathrm{low}}\right)$ and $w_i\left(x_{\mathrm{high}}\right)$, respectively. The evaluation curves $x_{\mathrm{low}}$ and $x_{\mathrm{high}}$ are highlighted, showing how the local neighborhood adapts to the functional geometry.
• Index-response plots:Figure 14 shows scatterplots of the scalar index $Z_i$ versus Fat and Protein. Points are colored according to the response-availability indicator ${\delta }_i$ and their size encodes the local weights at the corresponding evaluation curve. This reveals the MAR pattern and the impact of local weighting.
• Directional projections: Finally, Figure 15 reports, for each evaluation point and direction u, the empirical density of the projected responses $⟨Y_i,u⟩$ (distinguishing observed and missing responses) together with a vertical line at the corresponding projected geometric quantile $⟨{\widehat{\mathbb{Q}}}^{u,x},u⟩$.

Figure 12. Local conditional geometric quantiles of $(\mathrm{Fat},\mathrm{Protein})$ under MAR. Kernel estimators of the conditional geometric quantile $Q(u\mid x)$ are shown at two functional evaluation curves (low-fat and high-fat spectra) from the Tecator data. The background displays a bivariate kernel density estimate of the responses. Black markers indicate local weighted means, and arrows represent the directional shift from the local mean to $\widehat{Q}(u\mid x)$ for $u\in \{(1,0),(0,1),(1,1)/\sqrt{2}\}$. Solid curves are $95\%$ Wald-type confidence ellipses based on the estimated asymptotic precision matrix. Color distinguishes the conditioning curve: orange = high-fat, yellow = low-fat.

Figure 13. Functional neighborhoods induced by the $L^2$-distance and kernel weights around the low- and high-fat evaluation curves. Line transparency encodes the local weights $w_i\left(x\right)$. (a) Neighborhood around the low-fat curve. (b) Neighborhood around the high-fat curve.

Figure 14. Index-response plots under the MAR mechanism. Points are colored according to response availability and their size reflects the local weights at the corresponding evaluation curve. (a) Index vs. Fat. (b) Index vs. Protein.

Figure 15. Directional view of the conditional geometric quantiles. For each evaluation curve and direction u, the empirical density of the projections $⟨Y_i,u⟩$ is displayed together with the projected geometric quantile (dashed line).

Overall, Figure 12 provides a genuinely geometric view of how the functional covariate modulates the entire conditional law of $(\mathrm{Fat},\mathrm{Protein})$, beyond what can be captured by conditional means or covariance structures alone. The separation between the two local centers confirms a nontrivial effect of the spectrum on the first-order location of the response, while the heterogeneous lengths and orientations of the mean-to-quantile displacement vectors reveal pronounced direction-dependent departures from local elliptical symmetry. These displacements quantify how conditional tail thickness, skewness, and anisotropy evolve with the spectral profile, thereby exposing changes in the shape of $Y\mid X=x$ rather than mere shifts. The associated Wald-type ellipses, derived from the asymptotic theory under small-ball probabilities and MAR weighting, translate local functional sparsity and response incompleteness into inferential uncertainty, making explicit how the effective information content varies across regions of the functional covariate space. Taken together, the figure demonstrates that directional conditional geometric quantiles furnish a robust, shape-sensitive, and inferentially tractable description of multivariate conditional structure in functional settings, where heavy tails, asymmetry, and missing responses jointly undermine classical moment-based summaries.

6. Concluding Remarks

This paper investigates multivariate quantile regression in a setting where the response $Y\in {\mathbb{R}}^q$ ($q\ge 2$) is vector-valued and the covariate X is a functional random element taking values in an infinite-dimensional space. The principal objective is to introduce a novel kernel-based estimator of conditional geometric quantiles tailored to functional data that are strictly stationary and ergodic, while explicitly accommodating missing-at-random (MAR) mechanisms.

We begin by rigorously formulating the quantile regression problem in the presence of incomplete functional data, highlighting the dual challenges imposed by (i) the infinite-dimensional nature of the covariate and (ii) the statistical complexity of multivariate responses. Our construction of the estimator relies on nonparametric kernel smoothing, which offers both flexibility and computational tractability in capturing the conditional geometry of multivariate quantiles.

From a theoretical standpoint, we establish the strong consistency of the proposed estimator, derive its rate of convergence, and characterize its asymptotic distribution. These asymptotic results form the backbone for developing confidence regions and inference procedures within the multivariate quantile regression framework. Collectively, these contributions furnish a versatile methodological tool for analyzing functional data with multivariate responses, enriching the understanding of intricate dependence patterns between high-dimensional responses and infinite-dimensional covariates in diverse applied domains.

A natural but deliberately deferred question concerns the optimal choice of smoothing parameters that minimize the mean squared error (MSE) of the kernel estimator. While crucial to refining the method’s finite-sample performance, bandwidth selection is a technically distinct problem. We thus relegate to future work a comprehensive investigation of data-driven or asymptotically optimal bandwidth selectors, such as cross-validation or plug-in approaches, within the present functional and multivariate quantile context.

Beyond bandwidth selection, a central theoretical challenge in nonparametric functional data analysis (NFDA) is the bias inherent to classical kernel estimators. Although local linear smoothing provides a first-order bias reduction strategy, bias remains a dominant source of error when covariates are infinite-dimensional and responses are multivariate. Our computational experiments reveal that supplementing the kernel estimator with local linear bias-correction (CB) procedures substantially improves estimation accuracy. Nevertheless, rigorous mathematical foundations for such bias corrections—especially their asymptotic control and limit distributions—are still largely absent from the NFDA literature. To our knowledge, no exhaustive asymptotic treatment of bias-corrected kernel estimators has yet been undertaken in this functional setting. Addressing this theoretical gap represents an important and fertile avenue for future research.

In particular, the proposed framework could profitably integrate established bias-correction algorithms—such as those developed in [81,82]—to further enhance the trade-off between bias reduction and variance control. We anticipate that combining local linear smoothing with these CB techniques would markedly elevate estimation quality, thereby strengthening the robustness and efficiency of nonparametric functional quantile regression.

Another promising line of inquiry is the extension of our methodology to alternative nonparametric paradigms beyond kernel smoothing. Potential candidates include wavelet-based estimators [83,84], delta-sequence approaches [85], k-nearest neighbor (kNN) estimators [11], and other local polynomial methods [86]. These techniques, widely adopted in functional and high-dimensional statistics, may benefit from the bias-correction principles and asymptotic analysis developed here, opening the door to new synergies across nonparametric estimation frameworks.

Finally, it would be valuable to relax the stationarity assumption imposed throughout this study. Although strict stationarity simplifies asymptotic derivations, it may not be tenable in dynamic or evolving systems. Extending our theory to locally stationary or time-varying ergodic processes would enable the conditional quantile function to evolve smoothly over time, thereby accommodating nonstationary or transient data structures. Pursuing uniform limit theorems and related asymptotic results in such nonstationary contexts would constitute a natural and intellectually rewarding continuation of the present work.

7. Mathematical Developments

This section presents the proofs of our results and uses the notation introduced above.

Lemma 1 

([87]). Let ${\left(Z_n\right)}_{n\ge 1}$ be a sequence of real martingale differences with respect to the filtration ${\left({\mathcal{F}}_n\right)}_{n\ge 1}$, where ${\mathcal{F}}_n=\sigma (Z_1,\dots ,Z_n)$ is the σ-field generated by $Z_1,\dots ,Z_n$. Define $S_n={\sum }_{i=1}^n{Z}_i$. Assume that there exist a non-negative constant C and non-negative real numbers ${\left(d_n\right)}_{n\ge 1}$ such that for all $p\ge 2$ and all $n\ge 1$,

$$\mathbb{E}\left(Z_n^p|{\mathcal{F}}_{n-1}\right)\le C^{p-2}p!d_n^{2}a.s.$$

Then, for any $ϵ>0$, we have

$$\mathbb{P}\left(|S_n|>ϵ\right)\le 2exp\left(-{\displaystyle \frac{{ϵ}^2}{2(D_n+Cϵ)}}\right),$$

where $D_n={\sum }_{i=1}^n{d}_i^2$.

Let us introduce the following notation:

$${\overline{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)={\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left[{\varrho }_i{\mathcal{R}}^{j-1}\left(u;Y_i-m\right){\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right],\mathrm{for}j=1,2,$$

(16)

and

$${\overline{\mathcal{G}}}_{n,1}^x\left(0\right):={\overline{\mathcal{G}}}_{n,1}^x,$$

where $\mathbb{E}(X\mid \mathcal{F})$ denotes the conditional expectation of the random variable X given the $\sigma$-field $\mathcal{F}$. We define the conditional bias of ${\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)$ as

$${\mathcal{B}}_n^{u,x}\left(m\right)={\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\overline{\mathcal{G}}}_{n,1}^x}}-{\mathcal{G}}^{u,x}\left(m\right).$$

Consider now the following quantities:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_n^{u,x}\left(m\right)& =& -{\mathcal{B}}_n^{u,x}\left(m\right)\left({\widehat{\mathcal{G}}}_{n,1}^x-{\overline{\mathcal{G}}}_{n,1}^x\right),\hfill \end{array}$$

(17)

and

$$\begin{array}{ccc}\hfill {\mathcal{E}}_n^{u,x}\left(m\right)& =& \left({\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right)-{\mathcal{G}}^{u,x}\left(m\right)\left({\widehat{\mathcal{G}}}_{n,1}^x-{\overline{\mathcal{G}}}_{n,1}^x\right),\hfill \\ & =& {\mathcal{A}}_{n,2}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right){\mathcal{A}}_{n,1}\left(m\right),\hfill \end{array}$$

(18)

where ${\mathcal{A}}_{n,j}\left(m\right)={\widehat{\mathcal{G}}}_{n,j}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,j}^{u,x}\left(m\right)$, for $j=1,2$. It is then clear that the following decomposition holds:

$$\begin{array}{cc}\hfill {\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right)& ={\displaystyle \frac{{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\widehat{\mathcal{G}}}_{n,1}^x}}-{\mathcal{G}}^{u,x}\left(m\right)-{\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\overline{\mathcal{G}}}_{n,1}^x}}+{\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\overline{\mathcal{G}}}_{n,1}^x}}+{\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\widehat{\mathcal{G}}}_{n,1}^x}}\hfill \\ \hfill & -{\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\widehat{\mathcal{G}}}_{n,1}^x}}+{\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\widehat{\mathcal{G}}}_{n,1}^x}}{\mathcal{G}}^{u,x}\left(m\right)-{\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\widehat{\mathcal{G}}}_{n,1}^x}}{\mathcal{G}}^{u,x}\left(m\right)\hfill \\ \hfill & =\left({\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\overline{\mathcal{G}}}_{n,1}^x}}-{\mathcal{G}}^{u,x}\left(m\right)\right)-{\displaystyle \frac{1}{{\widehat{\mathcal{G}}}_{n,1}^x}}\left({\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)}{{\overline{\mathcal{G}}}_{n,1}^x}}-{\mathcal{G}}^{u,x}\left(m\right)\right)\times \left({\widehat{\mathcal{G}}}_{n,1}^x-{\overline{\mathcal{G}}}_{n,1}^x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{{\widehat{\mathcal{G}}}_{n,1}^x}}\left({\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right)-{\displaystyle \frac{{\mathcal{G}}^{u,x}\left(m\right)}{{\widehat{\mathcal{G}}}_{n,1}^x}}\left({\widehat{\mathcal{G}}}_{n,1}^x-{\overline{\mathcal{G}}}_{n,1}^x\right)\hfill \\ \hfill & ={\mathcal{B}}_n^{u,x}\left(m\right)+{\displaystyle \frac{{\mathcal{L}}_n^{u,x}\left(m\right)+{\mathcal{E}}_n^{u,x}\left(m\right)}{{\widehat{\mathcal{G}}}_{n,1}^x}}.\hfill \end{array}$$

(19)

Since ${\widehat{\mathcal{G}}}_{n,1}^x$ is independent of m, it follows from decomposition (19) that

$$\underset{m\in {\mathbb{R}}^q}{sup}\left|{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right)\right|\le \underset{m\in {\mathbb{R}}^q}{sup}\left|{\mathcal{B}}_n^{u,x}\left(m\right)\right|+{\displaystyle \frac{{sup}_{m\in {\mathbb{R}}^q}\left|{\mathcal{L}}_n^{u,x}\left(m\right)\right|+{sup}_{m\in {\mathbb{R}}^q}\left|{\mathcal{E}}_n^{u,x}\left(m\right)\right|}{{\widehat{\mathcal{G}}}_{n,1}^x}}.$$

(20)

The proof of Proposition 1 is split into several lemmas provided hereafter, establishing respectively the almost sure convergence of ${\widehat{\mathcal{G}}}_{n,1}^x$ to $p\left(x\right)$ and that of ${\mathcal{B}}_n^{u,x}\left(m\right)$, ${\mathcal{L}}_n^{u,x}\left(m\right)$, and ${\mathcal{E}}_n^{u,x}\left(m\right)$ (with rate) to zero. We start with the following technical lemma, whose proof can be found in [88].

Lemma 2.

Assume that conditions (A1) and (A2)(i), (A2)(ii), and (A2)(iv) hold true. For any real numbers $1\le j\le 2+\delta$ and $1\le k\le 2+\delta$ with $\delta >0$, as $n\to \infty$, we have

(i) 

${\displaystyle \frac{1}{\varphi \left(h\right)}\mathbb{E}\left[{\mathcal{K}}_i^j\left(x\right)\mid {\mathcal{F}}_{i-1}\right]={\mathcal{M}}_j{f}_{i,1}\left(x\right)+{\mathcal{O}}_{\mathrm{a}.\mathrm{s}.}\left(\frac{g_{i,x}\left(h\right)}{\varphi \left(h\right)}\right),}$

(ii) 

${\displaystyle \frac{1}{\varphi \left(h\right)}\mathbb{E}\left[{\mathcal{K}}_i^j\left(x\right)\right]={\mathcal{M}}_j{f}_1\left(x\right)+o\left(1\right),}$

(iii) 

${\displaystyle \frac{1}{{\varphi }^k\left(h\right)}{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^k={\mathcal{M}}_1^k{f}_1^k\left(x\right)+o\left(1\right),}$

where ${\mathcal{M}}_j$ is defined by

$${\displaystyle {\mathcal{M}}_j=K^j\left(1\right)-{\int }_0^1{\left(K^j\right)}^{\prime }\left(u\right){\tau }_0\left(u\right)du.}$$

Lemma 3.

Assume that hypotheses (A1)–(A2) and Conditions (10) are satisfied. Then, for any $x\in E$, we have

(i) 

${\widehat{\mathcal{G}}}_{n,1}^x-{\overline{\mathcal{G}}}_{n,1}^x={\mathcal{O}}_{\mathrm{a}.\mathrm{s}.}\left(\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right),$

(ii) 

${\displaystyle \underset{n\to \infty }{lim}{\widehat{\mathcal{G}}}_{n,1}^x=\underset{n\to \infty }{lim}{\overline{\mathcal{G}}}_{n,1}^x=p\left(x\right)\mathrm{a}.\mathrm{s}.,}$

(iii) 

$C_{n,1}^x\stackrel{\mathbb{P}}{⟶}0,\mathit{as}n\to \infty ,\mathit{and}\sqrt{n\varphi \left(h\right)}{C}_{n,1}^x\stackrel{\mathbb{P}}{⟶}0,\mathit{as}n\to \infty .$

Proof of Lemma 3.

By applying (5), we obtain the following decomposition:

$${\widehat{\mathcal{G}}}_{n,1}^x={\overline{\mathcal{G}}}_{n,1}^x+C_{n,1}^x,$$

(21)

where

$$C_{n,1}^x={\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n\left({\varrho }_i{\mathcal{K}}_i\left(x\right)-\mathbb{E}[{\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}]\right)$$

and

$${\overline{\mathcal{G}}}_{n,1}^x={\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n\mathbb{E}\left({\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right).$$

First, we need to establish the following convergence:

$${\overline{\mathcal{G}}}_{n,1}^x\stackrel{\mathbb{P}}{⟶}p\left(x\right),\mathrm{as}n\to \infty .$$

(22)

To achieve this, we utilize the properties of conditional expectations and the missing-at-random (MAR) mechanism. Under assumptions (A2:ii,iii), (A3:i), and (A9), and applying Lemma 2, we have the following detailed derivation. Intuitively, the conditional expectations of ${\varrho }_i{\mathcal{K}}_i\left(x\right)$ stabilize, and their averages align with $p\left(x\right)$ as n grows large:

$$\begin{array}{cc}\hfill {\overline{\mathcal{G}}}_{n,1}^x& ={\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left({\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left(\mathbb{E}\left({\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{A}}_{i-1}\right)\mid {\mathcal{F}}_{i-1}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left((p\left(x\right)+o\left(1\right)){\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right)\hfill \\ \hfill & =(p\left(x\right)+o\left(1\right)){\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}[{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}]\hfill \\ \hfill & =(p\left(x\right)+o\left(1\right)){\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\left(\varphi \left(h\right){\mathcal{M}}_1{f}_{i,1}\left(x\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}\left(h\right)\right)\right)\hfill \\ \hfill & =(p\left(x\right)+o\left(1\right)){\displaystyle \frac{\varphi \left(h\right)}{\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\left({\displaystyle \frac{1}{n}}\sum _{i=1}^n{\mathcal{M}}_1{f}_{i,1}\left(x\right)+{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\mathcal{O}}_{a.s.}\left({\displaystyle \frac{g_{i,x}\left(h\right)}{\varphi \left(h\right)}}\right)\right)\to p\left(x\right) \mathrm{a}\mathrm{s} n\to \infty ,\hfill \end{array}$$

confirming (22). Next, we show that the second component $C_{n,1}^x$ vanishes in probability as $n\to \infty$:

$$C_{n,1}^x\stackrel{\mathbb{P}}{⟶}0,\mathrm{as}n\to \infty .$$

(23)

To prove this, let us introduce the martingale differences

$${\mu }_{n,i}\left(x\right):={\varrho }_i{\mathcal{K}}_i\left(x\right)-\mathbb{E}[{\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}]$$

for $i=1,\dots ,n$. By construction, ${\left\{{\mu }_{n,i}\left(x\right)\right\}}_{i=1}^n$ forms a triangular array of martingale differences with respect to ${\mathcal{F}}_{i-1}$. Thus, we can rewrite

$$C_{n,1}^x={\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\mu }_{n,i}\left(x\right).$$

Applying Burkholder’s inequality [89,90,91] for martingale differences (cf. [92]), we obtain

$$\mathbb{P}\left(|C_{n,1}^x|>ϵ\right)=\mathbb{P}\left(\left|\sum _{i=1}^n{\mu }_{n,i}\left(x\right)\right|>ϵn\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]\right)\le C{\displaystyle \frac{\mathbb{E}\left[{\mu }_{n,1}^2\left(x\right)\right]}{{ϵ}^{2}n{\mathbb{E}}^2\left[{\mathcal{K}}_1\left(x\right)\right]}}.$$

Since ${\mu }_{n,1}^2\left(x\right)$ is bounded above by a term like ${\varrho }_1{\mathcal{K}}_1^2\left(x\right)$ and using the assumptions that guarantee $\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]>0$ and that $\mathbb{E}\left[{\varrho }_1{\mathcal{K}}_1^2\left(x\right)\right]$ is finite, we have

$$\mathbb{P}\left(|C_{n,1}^x|>ϵ\right)\le C{\displaystyle \frac{\mathbb{E}\left[{\varrho }_1{\mathcal{K}}_1^2\left(x\right)\right]}{{ϵ}^{2}n{\mathbb{E}}^2\left[{\mathcal{K}}_1\left(x\right)\right]}}\to 0\mathrm{as}n\to \infty .$$

This establishes (23). Therefore, using the same argument as used in (23), replacing $ϵ$ by $ϵ\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}$, we obtain (i) and (iii) of Lemma 3. Moreover, combining (21), (22), and (23), we conclude that Lemma 3(ii) follows immediately. □

In the following lemmas, we give the uniform asymptotic behavior of the conditional bias ${\mathcal{B}}_n^{u,x}\left(m\right)$ of ${\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)$ as well as that of ${\mathcal{L}}_n^{u,x}\left(m\right)$ and ${\mathcal{E}}_n^{u,x}\left(m\right)$ with respect to m.

Lemma 4.

Under conditions(A1), (A2), and (A3)(i) (with $j=0$), we have

$$\underset{m\in {\mathbb{R}}^q}{sup}\left|{\mathcal{B}}_n^{u,x}\left(m\right)\right|={\mathcal{O}}_{\mathrm{a}.\mathrm{s}.}\left(h^{\beta }\right).$$

(24)

If in addition Conditions (10) are satisfied, then we have

$$\underset{m\in {\mathbb{R}}^q}{sup}\left|{\mathcal{L}}_n^{u,x}\left(m\right)\right|={\mathcal{O}}_{\mathrm{a}.\mathrm{s}.}\left(h^{\beta }\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right).$$

(25)

Proof of Lemma 4.

Recall that

$${\mathcal{B}}_n^{u,x}\left(m\right)={\displaystyle \frac{{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right){\overline{\mathcal{G}}}_{n,1}^x}{{\overline{\mathcal{G}}}_{n,1}^x}}={\displaystyle \frac{B_n^{u,x}\left(m\right)}{{\overline{\mathcal{G}}}_{n,1}^x}}.$$

By the second part of Lemma 3, we have ${lim}_{n\to \infty }{\overline{\mathcal{G}}}_{n,1}^x=p\left(x\right)$ a.s. It suffices to prove that $B_n^{u,x}\left(m\right)={\mathcal{O}}_{\mathrm{a}.\mathrm{s}.}\left(h^{\beta }\right)$ as $n\to \infty$. Conditioning on the sigma-field ${\mathcal{A}}_{i-1}$ and using hypothesis (A4)(i) combined with Lemma 2, it follows from the definition (16) that

$$\begin{array}{cc}\hfill \left|B_n^{u,x}\left(m\right)\right|& =\left|{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right){\overline{\mathcal{G}}}_{n,1}^x\right|\hfill \\ \hfill & \le \left|{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right)\right|\hfill \\ \hfill & =\left|{\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n\mathbb{E}\left\{{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathbb{E}\left[\mathcal{R}\left(u;Y_i-m\right)-{\mathcal{G}}^{u,x}\left(m\right)\mid {\mathcal{A}}_{i-1}\right]\mid {\mathcal{F}}_{i-1}\right\}\right|\hfill \\ \hfill & =\left|{\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n\mathbb{E}\left\{{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathbb{E}\left[\mathcal{R}\left(u;Y_i-m\right)-{\mathcal{G}}^{u,x}\left(m\right)\mid X_i\right]\mid {\mathcal{F}}_{i-1}\right\}\right|\hfill \\ \hfill & =\left|{\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n\mathbb{E}\left[p\left(X_i\right){\mathcal{K}}_i\left(x\right)\left({\mathcal{G}}^{u,X_i}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right)\right)\right]\mid {\mathcal{F}}_{i-1}\right|\hfill \\ \hfill & \le \underset{x^{\prime }\in B(x,h)}{sup}\left|{\mathcal{G}}^{u,x^{\prime }}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right)\right|\left|{\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n\mathbb{E}\left[p\left(X_i\right){\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]\right|\hfill \\ \hfill & ={\mathcal{O}}_{\mathrm{a}.\mathrm{s}.}\left(h^{\beta }\right).\hfill \end{array}$$

The latter quantity results from hypothesis (A3)(i) with $j=0$ as the support of K is $[0,1]$. Now, to deal with the quantity ${\mathcal{L}}_n^{u,x}\left(m\right)$, recall that

$${\mathcal{L}}_n^{u,x}\left(m\right)=-{\mathcal{B}}_n^{u,x}\left(m\right)\left({\widehat{\mathcal{G}}}_{n,1}^x-{\overline{\mathcal{G}}}_{n,1}^x\right).$$

Therefore,

$$\underset{m\in {\mathbb{R}}^q}{sup}\left|{\mathcal{L}}_n^{u,x}\left(m\right)\right|=\left|{\widehat{\mathcal{G}}}_{n,1}^x-{\overline{\mathcal{G}}}_{n,1}^x\right|\underset{m\in {\mathbb{R}}^q}{sup}{\mathcal{B}}_n^{u,x}\left(m\right).$$

The statement (25) follows from (24) combined with the Lemma 3. □

Lemma 5.

Under assumptions $\left(\mathrm{A}1\right)$, $\left(\mathrm{A}2\right)$, $\left(\mathrm{A}3\right)\left(\mathrm{ii}\right)$, $\left(\mathrm{A}4\right)\left(\mathrm{i}\right)$, Conditions (15) and (16), we have

$$\underset{m\in {\mathbb{R}}^q}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|={\mathcal{O}}_{a.s.}\left(\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right).$$

Proof of Lemma 5.

Notice, for $j=1,2$, that we have

$${\widehat{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)-{\overline{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)={\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n{T}_{i,j}(m,x),$$

where

$$T_{i,j}(m,x)={\varrho }_i{\mathcal{K}}_i\left(x\right){\mathcal{R}}^{j-1}\left(u;Y_i-m\right)-\mathbb{E}\left[{\varrho }_i{\mathcal{K}}_i\left(x\right){\mathcal{R}}^{j-1}\left(u;Y_i-m\right)\mid {\mathcal{F}}_{i-1}\right].$$

Using Newton’s binomial formula, and for $j=2$, we obtain the following.

$$\begin{array}{c}{\displaystyle \mathbb{E}\left(T_{i,2}\left(x\right)\mid {\mathcal{F}}_{i-1}\right)}\hfill \\ {\displaystyle \hspace{1em}\hspace{1em}=\sum _{i=1}^n{C}_m^k\mathbb{E}\left({\left({\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m\right)\right)}^k\mid {\mathcal{F}}_{i-1}\right){(-1)}^{m-k}{\left[\mathbb{E}\left({\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m\right)\mid {\mathcal{F}}_{i-1}\right)\right]}^{m-k}}\hfill \end{array}$$

Making use of the Jensen inequality, one can write

$$\begin{array}{c}{\displaystyle \left|\mathbb{E}\left(T_{i,2}(m,x)\mid {\mathcal{F}}_{i-1}\right)\right|}\hfill \\ {\displaystyle \hspace{1em}\le \sum _{i=1}^n{C}_m^k\mathbb{E}\left({\left|{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m\right)\right|}^k\mid {\mathcal{F}}_{i-1}\right){\left[\mathbb{E}\left(\left|{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m\right)\right|\mid {\mathcal{F}}_{i-1}\right)\right]}^{m-k}}\\ {\displaystyle \hspace{1em}\le \sum _{i=1}^n{C}_m^k\mathbb{E}\left({\left|{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m\right)\right|}^k\mid {\mathcal{F}}_{i-1}\right)\mathbb{E}\left({\left|{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m\right)\right|}^{m-k}\mid {\mathcal{F}}_{i-1}\right).}\end{array}$$

For $u\in B^q$, we have

$$\begin{array}{cc}\hfill \left|\mathcal{R}\left(u;Y_i-m\right)\right|& =\left|\parallel Y_i-m\parallel +⟨u,Y-m⟩\right|\hfill \\ \hfill & \le \parallel Y_i-m\parallel +\parallel u\parallel \parallel Y_i-m\parallel \hfill \\ \hfill & \le 2\parallel Y_i-m\parallel ,\hfill \end{array}$$

for any $l\ge 2$

$$\begin{array}{c}{\displaystyle \mathbb{E}\left({\left|{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m\right)\right|}^l\mid {\mathcal{F}}_{i-1}\right)}\hfill \\ \hspace{1em}=\mathbb{E}\left(\mathbb{E}\left({\Delta }_i^l\left(x\right){\varrho }_i{\left|\mathcal{R}\left(u;Y_i-m\right)\right|}^l\mid {\mathcal{A}}_{i-1}\right)\mid {\mathcal{F}}_{i-1}\right)\hfill \\ \hspace{1em}=\mathbb{E}\left({\Delta }_i^l\left(x\right)\mathbb{E}\left({\varrho }_i{\left|\mathcal{R}\left(u;Y_i-m\right)\right|}^l\mid X_i\right)\mid {\mathcal{F}}_{i-1}\right)\hfill \\ \hspace{1em}\le 2^l\mathbb{E}\left(\left(\underset{u\in B(s,h)}{sup}\left|p\left(u\right)-p\left(x\right)\right|+p\left(x\right)\right)\left(\underset{v\in B(s,h)}{sup}\left|{\mathcal{Z}}_l\left(v\right)-{\mathcal{Z}}_l\left(x\right)\right|+{\mathcal{Z}}_l\left(x\right)\right){\Delta }_i^l\left(x\right)\mid {\mathcal{F}}_{i-1}\right)\hfill \\ \hspace{1em}\le C_2\mathbb{E}\left({\Delta }_i^l\left(x\right)\mid {\mathcal{F}}_{i-1}\right),\hfill \end{array}$$

where $C_2=2^l{C}_1$ is a positive constant. By Lemma 2, condition (A2)(ii), (A2)(iii) and K, ${\tau }_0$ bounded from above by $a_1$, $a_0$, respectively

$$\begin{array}{cc}\hfill \mathbb{E}\left({\Delta }_i^l\left(x\right)\mid {\mathcal{F}}_{i-1}\right)& =\varphi \left(h\right){\mathcal{M}}_l{f}_{i,1}\left(x\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}\left(h\right)\right)\hfill \\ \hfill & \le \varphi \left(h\right)a_1^l{f}_{i,1}\left(x\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}\left(h\right)\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{c}{\displaystyle \left|\mathbb{E}\left(T_{i,2}(m,x)\mid {\mathcal{F}}_{i-1}\right)\right|}\hfill \\ {\displaystyle \hspace{1em}\le C_2\sum _{i=1}^n{C}_m^k\left((\varphi \left(h\right)a_1^k{f}_{i,1}\left(x\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}\left(h\right)\right))(\varphi \left(h\right)a_1^{m-k}{f}_{i,1}\left(x\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}\left(h\right)\right))\right)}\hfill \\ {\displaystyle \hspace{1em}\le C_2\sum _{i=1}^n{C}_m^k\left({\varphi }^2\left(h\right)a_1^m{f}_{i,1}^2\left(x\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}^2\left(h\right)\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}\left(h\right)\right)\varphi \left(h\right)f_{i,1}\left(x\right)\left[a_1^k+a_1^{m-k}\right]\right)}\hfill \\ \hspace{1em}\le C_2{2}^m{\varphi }^2\left(h\right)a_1^m{f}_{i,1}^2\left(x\right)+2^m{\mathcal{O}}_{a.s.}\left(g_{i,x}^2\left(h\right)\right)+2{(a_1+1)}^m{\mathcal{O}}_{a.s.}\left(g_{i,x}\left(h\right)\right)\varphi \left(h\right)f_{i,1}\left(x\right)\hfill \\ \hspace{1em}\le {(2C_2^{1/m}max(1,a_1^2))}^m\left(\varphi \left(h\right)f_{i,1}\left(x\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}^2\left(h\right)\right)+{\mathcal{O}}_{a.s.}\left(g_{i,x}\left(h\right)\right)\varphi \left(h\right)f_{i,1}\left(x\right)\right)\hfill \\ \hspace{1em}\le C^{m-2}\varphi \left(h\right)\left(C_2{C}^2{b}_i\left(x\right)+o_{a.s.}\left(1\right)\right)\hfill \\ \hspace{1em}\le m!C^{m-2}\varphi \left(h\right)\left(C_2{C}^2{b}_i\left(x\right)+o_{a.s.}\left(1\right)\right),\hfill \end{array}$$

where $C=2max(1,a_1^2)$. The $m!$ term is both technically convenient and analytically powerful for establishing rigorous control over the tail behavior of unbounded martingale sums. It arises naturally from the Taylor expansion of the exponential function, which is used to bound the moment generating function, thereby leading to sub-Gaussian or sub-exponential upper bounds. In order to show almost sure convergence, we assume again $d_i^2=\varphi \left(h\right)\left(C_2{C}^2{b}_i\left(x\right)+o_{a.s.}\left(1\right)\right)$, ${\displaystyle D_n=\sum _i{d}_i^2}$, by Lemma 1 and observing (A2)(ii) and (A2)(v) that ${\displaystyle \frac{1}{n}}{D}_n=\varphi \left(h\right)\left(C_2{C}^2{b}_i\left(x\right)+o_{a.s.}\left(1\right)\right)\mathrm{as}n\to \infty$, $D_n=O\left(n\varphi \left(h\right)\right)$ and

$${\displaystyle S_n=\sum _{i=1}^n{T}_{i,2}(m,x).}$$

Thus, for any ${ϵ}_0>0$, we have

$$\begin{array}{c}{\displaystyle \mathbb{P}\left(|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)|>{ϵ}_0{\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right)}\hfill \\ \hspace{1em}=\mathbb{P}\left(\left|\sum _{i=1}^n{T}_{i,2}(m,x)\right|>{ϵ}_{0}n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]{\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right)\hfill \\ \hspace{1em}\le 2exp\left(-{\displaystyle \frac{n^2{ϵ}_0^2{\mathbb{E}}^2\left[{\mathcal{K}}_1\left(x\right)\right]\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}{2\left[\mathcal{O}\left(n\varphi \left(h\right)\right)+C{ϵ}_{0}n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]{\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right]}}\right)\hfill \\ \hspace{1em}\le 2exp\left(-{\displaystyle \frac{{\left(\mathcal{O}\left(n\varphi \right(h\left)\right)\right)}^2{ϵ}_0^2\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}{2\mathcal{O}\left(n\varphi \left(h\right)\right)\left[1+C{ϵ}_0{\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right]}}\right)\hfill \\ \hspace{1em}\le 2exp\left(-{\displaystyle \frac{C^{\prime }{ϵ}_0^{2}logn}{2\left[1+o\left(1\right)\right]}}\right)\hfill \\ \hspace{1em}\le 2exp\left(-C_1^{\prime }{ϵ}_0^{2}logn\right)\hfill \end{array}$$

(26)

$$\begin{array}{ccc}& \le & 2n^{-C_1^{\prime }{ϵ}_0^2},\hfill \end{array}$$

(27)

where $C_1^{\prime }$ denotes a strictly positive constant. Hence, by choosing ${ϵ}_0$ that is sufficiently large, we obtain

$${\displaystyle \sum _{n=1}^{\infty }\mathbb{P}\left(|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)|>{ϵ}_0{\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right)<\infty .}$$

(28)

If we take $Y_i={\mathbf{I}}_q$, for any $i\ge 1$ in the definitions ${\widehat{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)$ and ${\overline{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)$, we obtain under the same assumptions as in the previous lemma, except for hypotheses (A3)(ii) and (A4)(i),

$${\displaystyle \sum _{n=1}^{\infty }\mathbb{P}\left(|{\widehat{\mathcal{G}}}_{n,1}^x-{\overline{\mathcal{G}}}_{n,1}^x|>{ϵ}_0{\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right)<\infty .}$$

(29)

We conclude by the Borel–Cantelli Lemma. For $m\in {\mathbb{R}}^q$ and $r>0$, let

$$S(m,r)=\{m^{\prime }\in {\mathbb{R}}^q: \parallel m^{\prime }-m\parallel \le r\},$$

be the sphere of radius r centered at m. For $\gamma >1$, let ${\left[-n^{\gamma },n^{\gamma }\right]}^q$ be an interval of ${\mathbb{R}}^q$. Divide $\left[-n^{\gamma },n^{\gamma }\right]$ into $a_n$ subintervals each of length $l_n=\left[{\displaystyle \frac{2n^{\gamma }}{a_n^q}}\right]$ (where $\left[t\right]$ is the integer part of t). Since the set $S(0,n^{\gamma })=\{m^{\prime }\in {\mathbb{R}}^q: \parallel m^{\prime }\parallel \le n^{\gamma }\}$ is compact, it can be covered by $a_n^q$ bounded hypercubes of the form

$$S_{n,\rho }:=S(m_{\rho },l_n)=\{m^{\prime }\in {\mathbb{R}}^q: \parallel m^{\prime }-m_{\rho }\parallel \le l_n\},\rho =1,\dots ,a_n^q.$$

We have

$$\begin{array}{ccc}\hfill \underset{m\in {\mathbb{R}}^q}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|& \le & \underset{\parallel m\parallel >n^{\gamma }}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|+\underset{\parallel m\parallel \le n^{\gamma }}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|\hfill \\ & :=& I_1+I_2.\hfill \end{array}$$

(30)

The term $I_1$ of Equation (30) can be written as follows:

$$\begin{array}{ccc}{I}_1\hfill & =\hfill & \underset{\parallel m\parallel >n^{\gamma }}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|\hfill \\ \hfill & \le \hfill & \underset{\parallel m\parallel >n^{\gamma }}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|+\underset{\parallel m\parallel >n^{\gamma }}{sup}\left|{\mathcal{G}}^{u,x}\left(m\right)\right|+\underset{\parallel m\parallel >n^{\gamma }}{sup}\left|{\mathcal{G}}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|\hfill \end{array}$$

By Lemma 4 and the condition (10) is satisfied, we observe that

$$\begin{array}{cc}\hfill {\alpha }_n\left|{\mathcal{G}}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|& ={\mathcal{O}}_{a.s.}\left({\alpha }_n{\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right)\hfill \\ \hfill & ={\mathcal{O}}_{a.s.}\left(1\right).\hfill \end{array}$$

For the second term, whenever $\gamma >{\displaystyle \frac{1}{2}}$ and the condition (11) is satisfied, we have

$$\begin{array}{cc}\hfill {\alpha }_n\underset{\parallel m\parallel >n^{\gamma }}{sup}\left|{\mathcal{G}}^{u,x}\left(m\right)\right|& \le {\displaystyle \frac{{\alpha }_n}{n^{\gamma }}}\underset{\parallel m\parallel >n^{\gamma }}{sup}\parallel m\parallel \left|{\mathcal{G}}^{u,x}\left(m\right)\right|\hfill \\ \hfill & =o\left(1\right).\hfill \end{array}$$

In order to evaluate the first term, let us denote by

$${\Omega }_1:=\left\{i=\overline{1,n}:\parallel Y_i-m\parallel >{\displaystyle \frac{n^{\gamma }}{2}}\right\}$$

and

$${\Omega }_2:=\left\{i=\overline{1,n}:\parallel Y_i-m\parallel \le {\displaystyle \frac{n^{\gamma }}{2}}\right\},$$

we have for any $ϵ>0$

$$\begin{array}{cc}\hfill \mathbb{P}\left[{\alpha }_n\underset{\parallel m\parallel >n^{\gamma }}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|>ϵ\right]& =\mathbb{P}\left[{\alpha }_n\underset{\parallel m\parallel >n^{\gamma }}{sup}\left|{\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n{\varrho }_i\mathcal{R}\left(u;Y_i-m\right){\mathcal{K}}_i\left(x\right)\right|>ϵ\right]\hfill \\ \hfill & \le \mathbb{P}\left[{\alpha }_n\underset{\parallel m\parallel >n^{\gamma }}{sup}{\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)\left|\mathcal{R}\left(u;Y_i-m\right)\right|>ϵ\right]\hfill \\ \hfill & \le \mathbb{P}\left[{\displaystyle {\alpha }_n\underset{\parallel m\parallel >n^{\gamma }}{sup}\frac{2}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}\sum _{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)\parallel Y_i-m\parallel >ϵ}\right]\hfill \\ \hfill & \le \mathbb{P}\left(J_{n,1}\left(\omega \right)\right)+\mathbb{P}\left(J_{n,2}\left(\omega \right)\right),\hfill \end{array}$$

where

$${\displaystyle J_{n,1}\left(\omega \right)=\{\omega :{\alpha }_n\underset{\parallel m\parallel >n^{\gamma }}{sup}\frac{2}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}\sum _{{\Omega }_1}{\varrho }_i{\mathcal{K}}_i\left(x\right)\parallel Y_i-m\parallel >\frac{ϵ}{2}\},}$$

and

$${\displaystyle J_{n,2}\left(\omega \right)=\{\omega :{\alpha }_n\underset{\parallel m\parallel >n^{\gamma }}{sup}\frac{2}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}\sum _{{\Omega }_2}{\varrho }_i{\mathcal{K}}_i\left(x\right)\parallel Y_i-m\parallel >\frac{ϵ}{2}\}.}$$

The event $J_{n,1}\left(\omega \right)$ is nonempty if and only if there exists at least $i_0$: $1\le i_0\le n$ such that $\parallel Y_i-m\parallel >{\displaystyle \frac{n^{\gamma }}{2}}\}$. Hence,

$${\displaystyle \{J_{n,1}\left(\omega \right)\ne \varnothing \}\subset \bigcup _{i=1}^n\{\omega : \parallel Y_i-m\parallel >\frac{n^{\gamma }}{2}\},}$$

Applying Markov’s inequality and if $\mathbb{E}\left[\parallel Y_i-m\parallel \right]<\infty$, we obtain the following result:

$$\begin{array}{cc}\hfill \mathbb{P}\left(J_{n,1}\left(\omega \right)\ne \varnothing \right)& {\displaystyle \le \sum _{i=1}^n\mathbb{P}\left(\{\omega : \parallel Y_i-m\parallel >\frac{n^{\gamma }}{2}\}\right)}\hfill \\ \hfill & \le {\displaystyle \frac{n\mathbb{E}\left[\parallel Y_i-m\parallel \right]}{\frac{n^{\gamma }}{2}}}\le C{n}^{1-\gamma },\hfill \end{array}$$

for $\gamma >1$, we conclude by Borel–Cantelli’s Lemma that $J_{n,1}\left(\omega \right)=o_{a.s.}\left(1\right).$ For the second term $J_{n,2}\left(\omega \right)$, under set ${\Omega }_2$ we proceed similarly to $J_{n,1}\left(\omega \right)$, one may write

$$\parallel Y_{i_0}-m\parallel \le {\displaystyle \frac{n^{\gamma }}{2}}⟹\parallel m\parallel \ge \parallel Y_{i_0}-m\parallel .$$

Moreover, assuming the above conditions, the triangle inequality implies that

$${\displaystyle \frac{n^{\gamma }}{2}}<\parallel m\parallel -\parallel Y_{i_0}-m\parallel =\left|-\left(\parallel m\parallel -\parallel Y_{i_0}-m\parallel \right)\right|\le \parallel Y_{i_0}\parallel .$$

We then infer

$$\begin{array}{cc}\hfill \mathbb{P}\left(\{J_{n,2}\left(\omega \right)\ne \varnothing \}\right)& {\displaystyle \le \sum _{i=1}^n\mathbb{P}\left(\{\omega : \parallel Y_i-m\parallel \le \frac{n^{\gamma }}{2}\}\right)}\hfill \\ \hfill & {\displaystyle \le \sum _{i=1}^n\mathbb{P}\left(\parallel Y_{i_0}\parallel >\frac{n^{\gamma }}{2}\right),}\hfill \end{array}$$

applying Markov’s inequality and if $\mathbb{E}\left[\parallel Y_{i_0}\parallel \right]<\infty$, we obtain by Borel–Cantelli’s Lemma that

$$J_{n,2}\left(\omega \right)=o_{a.s.}\left(1\right).$$

The demonstration of $I_2$, for $j=1,2$, is based on the following inequality

$$\begin{array}{ccc}\hfill I_2& =& {\displaystyle \underset{\parallel m\parallel \le n^{\gamma }}{sup}\left|{\widehat{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)-{\overline{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)\right|}\hfill \\ & \le & {\displaystyle \underset{1\le \rho \le a_n^q}{max}\underset{m\in S_{n,\rho }}{sup}\left|{\widehat{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)-{\widehat{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m_{\rho }\right)\right|}\hfill \\ & & {\displaystyle +\underset{1\le \rho \le a_n^q}{max}\underset{m\in S_{n,\rho }}{sup}\left|{\widehat{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m_{\rho }\right)-{\overline{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m_{\rho }\right)\right|}\hfill \\ & & {\displaystyle +\underset{1\le \rho \le a_n^q}{max}\underset{m\in S_{n,\rho }}{sup}\left|{\overline{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m_{\rho }\right)-{\overline{\mathcal{G}}}_{n,j}^{(j-1)u,x}\left((j-1)m\right)\right|}\hfill \\ & :=& I_{n,1}\left(j\right)+I_{n,2}\left(j\right)+I_{n,3}\left(j\right).\hfill \end{array}$$

(31)

For $j=1$ in Equation (31), and by applying Lemma 3, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \underset{\parallel m\parallel \le n^{\gamma }}{sup}\left|{\widehat{\mathcal{G}}}_{n,1}^{,x}-{\overline{\mathcal{G}}}_{n,1}^x\right|}& =& {\displaystyle \underset{1\le \rho \le a_n^q}{max}\left|{\widehat{\mathcal{G}}}_{n,1}^{,x}-{\overline{\mathcal{G}}}_{n,1}^x\right|}\hfill \\ & =& \mathbb{P}\left(\left|C_{n,1}^x\right|>ϵ\right)⟶0,\mathrm{as} n⟶\infty .\hfill \end{array}$$

(32)

For $j=2$, $I_2$ is written as follows:

$$\begin{array}{ccc}\hfill I_2& =& {\displaystyle \underset{\parallel m\parallel \le n^{\gamma }}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|}\hfill \\ & \le & {\displaystyle \underset{1\le \rho \le a_n^q}{max}\underset{m\in S_{n,\rho }}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m_{\rho }\right)\right|}\hfill \\ & & {\displaystyle +\underset{1\le \rho \le a_n^q}{max}\underset{m\in S_{n,\rho }}{sup}\left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m_{\rho }\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m_{\rho }\right)\right|}\hfill \\ & & {\displaystyle +\underset{1\le \rho \le a_n^q}{max}\underset{m\in S_{n,\rho }}{sup}\left|{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m_{\rho }\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|}\hfill \\ & :=& I_{n,1}\left(2\right)+I_{n,2}\left(2\right)+I_{n,3}\left(2\right).\hfill \end{array}$$

(33)

For the second term $I_{n,2}\left(2\right)$, we will use the notation

$${\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m_{\rho }\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m_{\rho }\right)={\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n{T}_{i,2}(m_{\rho },x),$$

where

$$T_{i,2}(m_{\rho },x)={\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m_{\rho }\right)-\mathbb{E}\left[{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathcal{R}\left(u;Y_i-m_{\rho }\right)\mid {\mathcal{F}}_{i-1}\right].$$

Applying the exponential inequality (27), we therefore obtain that

$$\mathbb{P}\left(I_{n,2}\left(2\right)>{ϵ}_0{\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right)\le 2a_n^q{n}^{-C_1^{\prime }{ϵ}_0^2},$$

(34)

One may choose ${ϵ}_0$ large enough such that

$${\displaystyle \sum _n\mathbb{P}\left(I_{n,2}\left(2\right)>ϵ\right)<\infty .}$$

(35)

We now observe, for $I_{n,1}\left(2\right)$ and $I_{n,3}\left(2\right)$ that

$$\begin{array}{c}\hfill {\displaystyle \left|{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)-{\widehat{\mathcal{G}}}_{n,2}^{u,x}\left(m_{\rho }\right)\right|}\\ =& \left|{\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)\left(\mathcal{R}\left(u;Y_i-m\right)-\mathcal{R}\left(u;Y_i-m_{\rho }\right)\right)\right|\hfill \\ \le & {\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)\left(\left|\parallel Y_i-m\parallel -\parallel Y_i-m_{\rho }\parallel \right|+\left|⟨u,Y-m⟩-⟨u,Y-m_{\rho }⟩\right|\right)\hfill \\ \le & {\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)\left(\parallel m_{\rho }-m\parallel +\left|⟨u,m_{\rho }-m⟩\right|\right)\hfill \\ \le & {\displaystyle \frac{2}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)\parallel m_{\rho }-m\parallel \hfill \\ \le & 2{\widehat{\mathcal{G}}}_{n,1}^x{l}_n\hfill \end{array}$$

(36)

$$\begin{array}{ll}⟶\hfill & 2p\left(x\right)l_n,\mathrm{as} n⟶\infty .\hfill \end{array}$$

(37)

and

$$\begin{array}{c}\hfill {\displaystyle \left|{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m_{\rho }\right)-{\overline{\mathcal{G}}}_{n,2}^{u,x}\left(m\right)\right|}\\ =& \left|{\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n\mathbb{E}\left({\varrho }_i{\mathcal{K}}_i\left(x\right)\left(\mathcal{R}\left(u;Y_i-m_{\rho }\right)-\mathcal{R}\left(u;Y_i-m\right)\mid {\mathcal{F}}_{i-1}\right)\right)\right|\hfill \\ =& \left|{\displaystyle \frac{1}{n\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\sum _{i=1}^n\mathbb{E}\left[\mathbb{E}\left({\varrho }_i{\mathcal{K}}_i\left(x\right)\left(\mathcal{R}\left(u;Y_i-m_{\rho }\right)-\mathcal{R}\left(u;Y_i-m\right)\mid {\mathcal{A}}_{i-1}\right)\mid {\mathcal{F}}_{i-1}\right)\right]\right|\hfill \end{array}$$

(38)

$$\begin{array}{cc}\le & {\displaystyle {\overline{\mathcal{G}}}_{n,1}^x\underset{m\in S_{n,\rho }}{sup}\parallel {\mathcal{G}}^{u,X_i}\left(m_{\rho }\right)-{\mathcal{G}}^{u,X_i}\left(m\right)\parallel \le {\overline{\mathcal{G}}}_{n,1}^x{l}_n^{\delta }⟶p\left(x\right)l_n^{\delta },\mathrm{as} n⟶\infty .}\hfill \end{array}$$

(39)

Let ${\alpha }_n={\left({\displaystyle \frac{n\varphi \left(h\right)}{logn}}\right)}^{{\displaystyle \frac{1}{2}}}$ and by choosing $a_n^q=\left[{\alpha }_n{n}^{\gamma }logn\right]$

$$\begin{array}{ccc}\hfill {\alpha }_n\left(I_{n,1}\left(2\right)+I_{n,3}\left(2\right)\right)& =& \mathcal{O}\left({\alpha }_n\left(l_n+l_n^{\delta }\right)\right)\hfill \\ & =& \mathcal{O}\left({\displaystyle \frac{1}{{\left(n\varphi \left(h\right)\right)}^{\nu }logn}}\right)\hfill \\ & =& o\left(1\right),\hfill \end{array}$$

(40)

hence the result, whenever $\nu >0$. □

Lemma 6.

Under the same assumptions as in Lemma 5, we have

$$\underset{m\in {\mathbb{R}}^q}{sup}\left|{\mathcal{E}}_n^{u,x}\left(m\right)\right|={\mathcal{O}}_{a.s.}\left({\left({\displaystyle \frac{logn}{n\varphi \left(h\right)}}\right)}^{1/2}\right).$$

(41)

Proof of Lemma 6.

The proof of the statement (41) follows from the decomposition (18) and Lemmas 3, 5. □

Proof of Proposition 1.

The proof follows from the decomposition (20) and Lemmas 3–6. □

Proof of Theorem 1.

From the definitions of ${\mathbb{Q}}^{u,x}$ and ${\widehat{\mathbb{Q}}}_n^{u,x}$, and the existence and uniqueness of these quantities, we have, for any $x\in E$ and $u\in B^q$,

$${\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)=\underset{m\in {\mathbb{R}}^q}{inf}{\mathcal{G}}^{u,x}\left(m\right)\mathrm{and}{\widehat{\mathcal{G}}}_n^{u,x}\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)=\underset{m\in {\mathbb{R}}^q}{inf}{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right).$$

(42)

Using similar arguments as in the proof of Theorem 3.1 of [59], we can write

$$\left|{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)-{\mathcal{G}}^{u,x}\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)\right|\le 2\underset{m\in {\mathbb{R}}^q}{sup}\left|{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right)\right|.$$

(43)

Moreover, since for any fixed $x\in E$ and $u\in B^q$, the function ${\mathcal{G}}^{u,x}\left(·\right)$ is uniformly continuous, and because ${\mathbb{Q}}^{u,x}$ is the unique minimizer of ${\mathcal{G}}^{u,x}\left(·\right)$, we have that for any $ϵ>0$,

$$\underset{m:\parallel {\mathbb{Q}}^{u,x}-m\parallel \ge ϵ}{inf}{\mathcal{G}}^{u,x}\left(m\right)>{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right).$$

(44)

This implies that there exists, for any $ϵ>0$, a number $\eta \left(ϵ\right)>0$ such that, for every m with $\parallel {\mathbb{Q}}^{u,x}-m\parallel \ge ϵ$, we have

$${\mathcal{G}}^{u,x}\left(m\right)>{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)+\eta \left(ϵ\right).$$

(45)

It follows from inequality (43) that, for any $x\in E$ and $m\in B^q$,

$$\begin{array}{cc}\hfill \sum _{n\ge 1}\mathbb{P}\left(∥{\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}∥>ϵ\right)& \le \sum _{n\ge 1}\mathbb{P}\left({\mathcal{G}}^{u,x}\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)>{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)+\eta \left(ϵ\right)\right)\hfill \\ \hfill & \le \sum _{n\ge 1}\mathbb{P}\left(\underset{m\in {\mathbb{R}}^q}{sup}\left|{\widehat{\mathcal{G}}}_n^{u,x}\left(m\right)-{\mathcal{G}}^{u,x}\left(m\right)\right|>{\displaystyle \frac{\eta \left(ϵ\right)}{2}}\right)<\infty .\hfill \end{array}$$

(46)

Similarly to the proof of Proposition 1, the statement (12) then follows from an application of the Borel–Cantelli Lemma. □

Proof of Proposition 2.

In order to prove Proposition 2, we need to prove the following lemmas. Write

$$∥{\widehat{\mathbb{M}}}_n^x\left({\xi }_n\left(j\right)\right)-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥\le ∥{\widehat{\mathbb{M}}}_n^x\left({\xi }_n\left(j\right)\right)-{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)∥+∥{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥.$$

(47)

□

Lemma 7.

Under the same assumptions as in Proposition 2, we have

$$∥{\widehat{\mathbb{M}}}_n^x\left({\xi }_n\left(j\right)\right)-{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)∥\le C_k{B}_{k,n}\left(x\right),$$

(48)

where, for $k=1,2$, $C_k$ positive constants

$$B_{k,n}\left(x\right)={\displaystyle \frac{o_{a.s.}\left(1\right)}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{∥Y_i-{\mathbb{Q}}^{u,x}∥}^2}},$$

and

$$B_{k,n}\left(x\right)=o_{a.s.}\left(1\right).$$

(49)

Lemma 8.

Under the same assumptions as in Lemmas 3 and 5, using assumption (A6) and Lemma 1

$$∥{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)-\mathbb{E}\left({\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right)∥=o_{a.s.}\left(\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right),$$

(50)

and if $\parallel {\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\parallel < \infty$ a.s., we get

$$∥\mathbb{E}\left({\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right)-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥=o_{a.s.}\left(1\right).$$

(51)

Proof of Lemma 7.

Let us define the following quantity $\mathcal{L}\left(y,m\right)=\mathcal{D}(y-m){\mathcal{D}}^{\top }(y-m)$ and observe that

$$\begin{array}{c}{\displaystyle ∥{\widehat{\mathbb{M}}}_n^x\left({\xi }_n\left(j\right)\right)-{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)∥}\hfill \\ \le & {\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n∥\mathbb{L}\left(Y_i,{\xi }_n\left(j\right)\right)-\mathbb{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right)∥{\varrho }_i{\mathcal{K}}_i\left(x\right)\hfill \\ \le & {\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\varrho }_i{\mathcal{K}}_i\left(x\right)\left(\parallel {\displaystyle \frac{1}{\parallel Y_i-{\xi }_n\left(j\right)\parallel }}\left({\mathbf{I}}_q-\mathcal{D}(Y_i-{\xi }_n\left(j\right)){\mathcal{D}}^{\top }(Y_i-{\xi }_n\left(j\right))\right)\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{\parallel Y_i-{\mathbb{Q}}^{u,x}\parallel }}\left({\mathbf{I}}_q-\mathcal{D}(Y_i-{\mathbb{Q}}^{u,x}){\mathcal{D}}^{\top }(Y_i-{\mathbb{Q}}^{u,x})\right)∥\right)\hfill \\ :=& {\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)\left|∥Y_i-{\mathbb{Q}}^{u,x}∥-∥Y_i-{\xi }_n\left(j\right)∥\right|}{∥Y_i-{\mathbb{Q}}^{u,x}∥∥Y_i-{\xi }_n\left(j\right)∥}}\hfill \\ \hfill & +{\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)∥∥Y_i-{\xi }_n\left(j\right)∥\mathcal{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right)-∥Y_i-{\mathbb{Q}}^{u,x}∥\mathcal{L}\left(Y_i,{\xi }_n\left(j\right)\right)∥}{∥Y_i-{\mathbb{Q}}^{u,x}∥∥Y_i-{\xi }_n\left(j\right)∥}}\hfill \\ \le & J_{1,n}+J_{2,n}.\hfill \end{array}$$

(52)

Using the triangle inequality, we can easily see that

$$J_{1,n}\le {\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)∥{\xi }_n\left(j\right)-{\mathbb{Q}}^{u,x}∥}{∥Y_i-{\mathbb{Q}}^{u,x}∥∥Y_i-{\xi }_n\left(j\right)∥}}.$$

(53)

Using Theorem 1, $∥{\xi }_n\left(j\right)-{\mathbb{Q}}^{u,x}∥\le ∥{\mathbb{Q}}_n^{u,x}-{\mathbb{Q}}^{u,x}∥$ and the fact that

$$∥Y_i-{\xi }_n\left(j\right)∥\le ∥Y_i-{\mathbb{Q}}^{u,x}∥.$$

This, implies that

$$J_{1,n}\le {\displaystyle \frac{o_{a.s.}\left(1\right)}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{∥Y_i-{\mathbb{Q}}^{u,x}∥}^2}}.$$

(54)

By adding and subtracting $∥Y_i-{\mathbb{Q}}^{u,x}∥\mathcal{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right)$, applying the triangle inequality, and using the fact that $∥\mathcal{D}\left(Y_i-m\right)∥=1$, we can write

$$\begin{array}{c}{\displaystyle ∥∥Y_i-{\xi }_n\left(j\right)∥\mathcal{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right)-∥Y_i-{\mathbb{Q}}^{u,x}∥\mathcal{L}\left(Y_i,{\xi }_n\left(j\right)\right)∥}\hfill \\ \hspace{1em}\le \left|∥Y_i-{\xi }_n\left(j\right)∥-∥Y_i-{\mathbb{Q}}^{u,x}∥\right|∥\mathcal{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right)∥\hfill \\ \hspace{1em}\hspace{1em}+∥Y_i-{\mathbb{Q}}^{u,x}∥∥\mathcal{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right)-\mathcal{L}\left(Y_i,{\xi }_n\left(j\right)\right)∥\hfill \\ \hspace{1em}\le ∥{\xi }_n\left(j\right)-{\mathbb{Q}}^{u,x}∥+∥Y_i-{\mathbb{Q}}^{u,x}∥∥\mathcal{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right)-\mathcal{L}\left(Y_i,{\xi }_n\left(j\right)\right)∥.\hfill \end{array}$$

(55)

Now observe that

$$\begin{array}{cc}\hfill \mathcal{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right)-\mathcal{L}\left(Y_i,{\xi }_n\left(j\right)\right)& =\left[\mathcal{D}\left(Y_i-{\mathbb{Q}}^{u,x}\right)-\mathcal{D}\left(Y_i-{\xi }_n\left(j\right)\right)\right]{\mathcal{D}}^{\top }\left(Y_i-{\mathbb{Q}}^{u,x}\right)\hfill \\ \hfill & +\mathcal{D}\left(Y_i-{\xi }_n\left(j\right)\right)\left[{\mathcal{D}}^{\top }\left(Y_i-{\mathbb{Q}}^{u,x}\right)-{\mathcal{D}}^{\top }\left(Y_i-{\xi }_n\left(j\right)\right)\right].\hfill \end{array}$$

(56)

Moreover, by adding and subtracting $∥Y_i-{\xi }_n\left(j\right)∥\mathcal{L}\left(Y_i,{\xi }_n\left(j\right)\right)$

$$\begin{array}{c}\hfill {\displaystyle \mathcal{D}\left(Y_i-{\mathbb{Q}}^{u,x}\right)-\mathcal{D}\left(Y_i-{\xi }_n\left(j\right)\right)}\\ =& {\displaystyle \frac{\left(Y_i-{\xi }_n\left(j\right)\right)\left[∥Y_i-{\xi }_n\left(j\right)∥-∥Y_i-{\mathbb{Q}}^{u,x}∥\right]+\left({\xi }_n\left(j\right)-{\mathbb{Q}}^{u,x}\right)∥Y_i-{\xi }_n\left(j\right)∥}{∥Y_i-{\mathbb{Q}}^{u,x}∥∥Y_i-{\xi }_n\left(j\right)∥}}.\hfill \end{array}$$

(57)

Then, by the triangle inequality,

$$∥\mathcal{D}\left(Y_i-{\mathbb{Q}}^{u,x}\right)-\mathcal{D}\left(Y_i-{\xi }_n\left(j\right)\right)∥\le {\displaystyle \frac{4∥{\xi }_n\left(j\right)-{\mathbb{Q}}^{u,x}∥}{∥Y_i-{\mathbb{Q}}^{u,x}∥}}.$$

(58)

The same result can be obtained for the second term in (56). Combining (55), (56), and (58), we get

$$J_{2,n}\le {\displaystyle \frac{5\times o_{a.s.}\left(1\right)}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{∥Y_i-{\mathbb{Q}}^{u,x}∥}^2}}.$$

(59)

The term

$${\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{∥Y_i-{\mathbb{Q}}^{u,x}∥}^2}}:={\widehat{{\mathcal{G}}^{\prime x}}}_{n,1},$$

which appears in (54) and (59) is computed similarly to Lemma 3, based on the following decomposition:

$${\widehat{{\mathcal{G}}^{\prime x}}}_{n,1}={\overline{{\mathcal{G}}^{\prime x}}}_{n,1}+C_{n,1}^{\prime x},$$

(60)

where

$$C_{n,1}^{\prime x}={\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n\left({\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{∥Y_i-{\mathbb{Q}}^{u,x}∥}^2}}-\mathbb{E}[{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{∥Y_i-{\mathbb{Q}}^{u,x}∥}^2}}\mid {\mathcal{F}}_{i-1}]\right)$$

and

$${\overline{{\mathcal{G}}^{\prime x}}}_{n,1}={\displaystyle \frac{1}{n\mathbb{E}{\mathcal{K}}_1\left(x\right)}}\sum _{i=1}^n\mathbb{E}\left({\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{∥Y_i-{\mathbb{Q}}^{u,x}∥}^2}}\mid {\mathcal{F}}_{i-1}\right).$$

Then, by (A3:ii), we can easily prove that

$${\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{{∥Y_i-{\mathbb{Q}}^{u,x}∥}^2}}=O_{\mathbb{P}}\left(1\right).$$

Finally, we get $B_{k,n}\left(x\right)=o_{a.s.}\left(1\right)$, for $k=1,2$, which allows us to conclude that

$$∥{\widehat{M}}_n^x\left({\xi }_n\left(j\right)\right)-{\widehat{M}}_n^x\left({\mathbb{Q}}^{u,x}\right)∥=o_{a.s.}\left(1\right).$$

(61)

Now we are interested in the second term on the right-hand side of (47). □

Proof of Lemma 8.

Write

$$\begin{array}{ccc}\hfill {\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)-{\mathbb{M}}^x\left(Q^{u,x}\right)& =& \underset{{\mathcal{K}}_{n,1}}{\underbrace{{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)-\mathbb{E}\left[{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right]}}\hfill \\ & & +\underset{{\mathcal{K}}_{n,2}}{\underbrace{\mathbb{E}\left[{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right]-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)}}.\hfill \end{array}$$

(62)

We have to show that each term ${\mathcal{K}}_{n,i}$ ($i=1,2$) is asymptotically negligible. We have

$${∥{\mathcal{K}}_{n,1}∥}^2=tr\left({\mathcal{K}}_{n,1}^T{\mathcal{K}}_{n,1}\right)=\sum _{k=1}^q\sum _{j=1}^q{\mathbb{Z}}_{k,j}^2,$$

(63)

where ${\left({\mathbb{Z}}_{k,j}\right)}_{1\le k,j\le q}$ is the general term of the matrix ${\mathcal{K}}_{n,1}^{\top }{\mathcal{K}}_{n,1}$, which can be written as

$$\begin{array}{cc}\hfill {\mathbb{Z}}_{k,j}& ={\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\left[{\mathbb{L}}_{k,j}\left(Y_i,{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)-\mathbb{E}\left({\mathbb{L}}_{k,j}\left(Y_i,{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n{M}_{n,i}\left(x,{\mathbb{Q}}^{u,x}\right),\hfill \end{array}$$

(64)

where

$$M_{n,i}\left(x,{\mathbb{Q}}^{u,x}\right):={\mathbb{L}}_{k,j}\left(Y_i,{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)-\mathbb{E}\left({\mathbb{L}}_{k,j}\left(Y_i,{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right)$$

is a martingale difference. Therefore, we can use Lemma 1 to obtain an exponential upper bound for the quantity ${\mathbb{Z}}_{k,j}$. Let us now check the conditions under which one can obtain the mentioned exponential upper bound. Following similar steps as in the proof of Lemma 5 and using assumption $\left(\mathrm{A}5\right)$, one can easily find that, for all $1\le k,j\le q$,

$${\mathbb{Z}}_{k,j}=o_{a.s.}\left(\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right).$$

(65)

To deal with the term ${\mathcal{K}}_{n,2}$, we have, by adding and subtracting $∥{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥{\overline{\mathcal{G}}}_{n,1}^x$ and the triangle inequality

$$\begin{array}{cc}\hfill ∥{\mathcal{K}}_{n,2}∥& =∥{\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left[\mathbb{L}\left(Y_i,{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥\hfill \\ \hfill & =∥{\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left[{\mathbb{M}}^{X_i}\left({\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥\hfill \\ \hfill & \le \underset{x^{\prime }\in B(x,h)}{sup}∥{\mathbb{M}}^{x^{\prime }}\left({\mathbb{Q}}^{u,x}\right)-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥\times {\overline{\mathcal{G}}}_{n,1}^x+∥{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥\times \left|{\overline{\mathcal{G}}}_{n,1}^x-1\right|,\hfill \end{array}$$

(66)

where ${\overline{\mathcal{G}}}_{n,1}^x$ is an appropriate normalizing constant. Therefore, using assumption $\left(\mathrm{A}6\right)$, Lemma 3, and the fact that the assumption (A3:ii) is satisfied, the norm of ${\mathbb{M}}^x\left(Q^{u,x}\right)$ is bounded, we get

$$∥{\mathcal{K}}_{n,2}∥=o_{a.s.}\left(1\right).$$

(67)

Finally, by combining (65) and (67), we get

$$∥{\widehat{\mathbb{M}}}_n^x\left({\mathbb{Q}}^{u,x}\right)-{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)∥=o_{a.s.}\left(1\right),$$

(68)

which concludes the proof of Proposition 2. □

Proof of Theorem 2.

Firstly, we introduce the following notation:

$$\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}:=-{\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left[{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right],$$

and let us denote $\Xi :=-{\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]}^{-1}$. To find the rate of strong consistency of our estimator ${\widehat{\mathbb{Q}}}_n^{u,x}$, we decompose ${\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}$ into the sum of two components. It follows from Proposition 2 and decomposition (13) that

$$\begin{array}{ccc}\hfill {\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}& =& \Xi \times {\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)\hfill \\ & =& \Xi \times \left[{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}+\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}\right].\hfill \end{array}$$

(69)

□

Lemma 9.

Under the assumptions of Lemma 2, we have

$$∥\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥={\mathcal{O}}_{a.s.}\left(\varphi \left(h\right)\right).$$

Proof of Lemma 9.

For $u\in B^q$, we observe that

$$\begin{array}{cc}\hfill \parallel \mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right)\parallel & =\parallel \mathcal{D}\left(Y_i-{\mathbb{Q}}^{u,x}\right)+u\parallel \hfill \\ \hfill & \le \parallel \mathcal{D}\left(Y_i-{\mathbb{Q}}^{u,x}\right)\parallel +\parallel u\parallel \hfill \\ \hfill & \le 2.\hfill \end{array}$$

Therefore,

$$∥\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥\le {\displaystyle \frac{2\left(p\right(x)+o(1\left)\right)}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left[{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right].$$

Since the kernel function $K\left(·\right)$ is bounded below by some constant $c^{\prime }$ (by assumption $\left(\mathrm{A}1\right)$), we have

$$\begin{array}{cc}\hfill ∥\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥& \le {\displaystyle \frac{2c^{\prime }(p\left(x\right)+o\left(1\right))}{n}}\sum _{i=1}^n\mathbb{E}\left[{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]\hfill \\ \hfill & {\displaystyle \le \left(2c^{\prime }(p\left(x\right)+o\left(1\right))\right){\mathcal{O}}_{a.s.}\left(\varphi \left(h\right)\right){\mathcal{M}}_1\frac{1}{n}\sum _{i=1}^n{f}_{i,1}\left(x\right)+\frac{1}{n}\sum _{i=1}^n{g}_{i,x}\left(h\right){\mathcal{O}}_{a.s.}\left(1\right).}\hfill \end{array}$$

Using Lemma 2 and assumptions (A2)(ii)–(iii), we conclude that

$$∥\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥={\mathcal{O}}_{a.s.}\left(\varphi \left(h\right)\right),$$

which completes the proof of Lemma 9. □

Lemma 10.

Under the assumptions of Lemma 3 and condition $\left(\mathrm{A}7\right)$, we have

$$∥{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥={\mathcal{O}}_{a.s.}\left(\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right).$$

Proof of Lemma 10.

Notice that we have

$$\begin{array}{c}\hfill {\displaystyle {\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}}\\ =& -{\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\left(\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)-\mathbb{E}\left[\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]\right),\hfill \end{array}$$

and let us denote

$$S_n:=\sum _{i=1}^n\left(\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)-\mathbb{E}\left[\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]\right).$$

Then,

$$∥{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥\le {\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}∥S_n∥.$$

To bound $∥S_n∥$, we will use the Azuma–Hoeffding [93,94] inequality for martingale differences. Note that

$$∥\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right)∥\le 2,$$

and ${\mathcal{K}}_i\left(x\right)$ is bounded by $\delta$, where $\delta$ is the bound on the kernel function K. Therefore, the differences

$$D_i:=\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)-\mathbb{E}\left[\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]$$

are bounded in norm by $2\delta$. Using the martingale convergence theorem and applying the Azuma–Hoeffding inequality, we get

$$\mathbb{P}\left(∥S_n∥>t\right)\le 2exp\left(-{\displaystyle \frac{t^2}{2n{\left(2\delta \right)}^2}}\right).$$

Setting $t=\sqrt{2n{\left(2\delta \right)}^{2}logn}$, we obtain

$$∥S_n∥={\mathcal{O}}_{a.s.}\left(\sqrt{nlogn}\right).$$

Thus,

$$∥{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥\le {\displaystyle \frac{1}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}{\mathcal{O}}_{a.s.}\left(\sqrt{nlogn}\right)={\mathcal{O}}_{a.s.}\left({\displaystyle \frac{\sqrt{logn}}{\sqrt{n}\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\right).$$

Using Lemma 2 and the fact that $\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]=\mathcal{O}\left(\varphi \left(h\right)\right)$, we have

$$∥{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥={\mathcal{O}}_{a.s.}\left(\sqrt{{\displaystyle \frac{logn}{n\varphi \left(h\right)}}}\right),$$

which completes the proof of Lemma 10. By combining Lemmas 9 and 10. Then,

$$∥{\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}∥\le \parallel \Xi \parallel \times \left[∥{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥+∥\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}∥\right].$$

Finally, because $\parallel \Xi \parallel <\infty$ and by Lemmas 9 and 10, we get the result of Theorem 2. □

Proof of Theorem 3.

Using the decomposition (69), we get under the assumptions of Proposition 2 that

$$\sqrt{n\varphi \left(h\right)}\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right)=\Xi \left(\sqrt{n\varphi \left(h\right)}\left[{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}\right]+{\mathcal{I}}_n^{\mathbb{Q}}\left(x\right)\right)$$

with

$${\mathcal{I}}_n^{\mathbb{Q}}\left(x\right)={\left(n\varphi \left(h\right)\right)}^{{\displaystyle \frac{1}{2}}}\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}$$

To prove Theorem 3 it suffices to show that $\sqrt{n\varphi \left(h\right)}\left[{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}\right]$ is asymptotically distributed as ${\mathcal{N}}_d\left(0,{\Sigma }^x\left({\mathbb{Q}}^{u,x}\right)\right)$, where ${\Sigma }^x\left({\mathbb{Q}}^{u,x}\right)$ is defined in Theorem 3 and that ${\mathcal{I}}_n^{\mathbb{Q}}\left(x\right)$ is asymptotically negligible in the following Lemmas. □

Lemma 11.

Under the assumptions (A1)–(A2), (A3)(i) with $(j=1)$, (A4)(ii) and $h^{\beta }\sqrt{n\varphi \left(h\right)}⟶0$ as $n⟶\infty$

• we have

$${\mathcal{I}}_n^{\mathbb{Q}}\left(x\right)=\mathcal{O}\left(h^{\beta }\sqrt{n\varphi \left(h\right)}\right).$$

Proof of Lemma 11.

Observe that, by double conditioning with respect to ${\mathcal{A}}_{i-1}$ and using assumptions (A4)(ii), one obtains

$$\begin{array}{ccc}\hfill {\mathcal{I}}_n^{\mathbb{Q}}\left(x\right)& =& -{\displaystyle \frac{\sqrt{n\varphi \left(h\right)}}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left[{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right]\hfill \\ & =& -{\displaystyle \frac{\sqrt{n\varphi \left(h\right)}}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left(\mathbb{E}\left[{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right)\mid {\mathcal{A}}_{i-1}\right]\mid {\mathcal{F}}_{i-1}\right)\hfill \\ & =& -{\displaystyle \frac{\sqrt{n\varphi \left(h\right)}}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left(\mathbb{E}\left[{\varrho }_i{\mathcal{K}}_i\left(x\right)\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right)\mid X_i\right]\mid {\mathcal{F}}_{i-1}\right)\hfill \\ & =& {\displaystyle \frac{\sqrt{n\varphi \left(h\right)}}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left({\nabla }_m{\mathcal{G}}^{u,X_i}\left({\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right)\hfill \\ & \le & \left({\nabla }_m{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)+\underset{x^{\prime }\in B(x,h)}{sup}∥{\nabla }_m{\mathcal{G}}^{u,x^{\prime }}\left({\mathbb{Q}}^{u,x}\right)-{\nabla }_m{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)∥\right)\hfill \\ & & \times {\displaystyle \frac{\sqrt{n\varphi \left(h\right)}}{n\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\sum _{i=1}^n\mathbb{E}\left({\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right).\hfill \end{array}$$

Using the fact that the conditional geometric quantile ${\mathbb{Q}}^{u,x}$ satisfies the following equation

$${\nabla }_m{\mathcal{G}}^{u,x}\left(m\right)=0,$$

(A3)(i) ($j=1$) and by Lemma 2, we readily obtain

$$\begin{array}{ccc}\hfill {\mathcal{I}}_n^{\mathbb{Q}}\left(x\right)& \le & \sqrt{n\varphi \left(h\right)}\times \underset{x^{\prime }\in B(x,h)}{sup}∥{\nabla }_m{\mathcal{G}}^{u,x^{\prime }}\left({\mathbb{Q}}^{u,x}\right)-{\nabla }_m{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)∥\hfill \\ & & \times \left(p\left(x\right)+o\left(1\right)\right){\displaystyle \frac{\varphi \left(h\right)}{\mathbb{E}\left[{\mathcal{K}}_1\left(x\right)\right]}}\left({\displaystyle \frac{1}{n}}\sum _{i=1}^n{\mathcal{M}}_1{f}_{i,1}\left(x\right)+{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\mathcal{O}}_{a.s.}\left({\displaystyle \frac{g_{i,x}\left(h\right)}{\varphi \left(h\right)}}\right)\right)\hfill \\ & =& \mathcal{O}\left(h^{\beta }\sqrt{n\varphi \left(h\right)}\right).\hfill \end{array}$$

Finally, the proof of Theorem 3 is achieved by according to the second lemma. □

Lemma 12.

Assume that the conditions (A1) and (A2)(i), (A2)(ii) and (A2)(iv), (A3)(i) (with $j=1$), (A4)(ii) and (A)(8) hold true. Then, we have

$$\sqrt{n\varphi \left(h\right)}\left[{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}\right]\stackrel{\mathcal{D}}{\to }{\mathcal{N}}_d\left(0,{\Sigma }^x\left({\mathbb{Q}}^{u,x}\right)\right),$$

where ${\Sigma }^x\left({\mathbb{Q}}^{u,x}\right)$ is the limiting covariance matrix of $\left[{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}\right]$.

Proof of Lemma 12.

Let us denote

$$H_{ni}=-\sqrt{{\displaystyle \frac{\varphi \left(h\right)}{n}}}\times \mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){\displaystyle \frac{{\varrho }_i{\mathcal{K}}_i\left(x\right)}{\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\mathrm{and}{\overline{H}}_{ni}=\mathbb{E}\left(H_{ni}\mid {\mathcal{F}}_{i-1}\right).$$

Then

$$\sqrt{n\varphi \left(h\right)}\left[{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)-\overline{{\nabla }_m{\widehat{\mathcal{G}}}_n^{u,x}\left({\mathbb{Q}}^{u,x}\right)}\right]=\sum _{i=1}^n\left(H_{ni}-{\overline{H}}_{ni}\right):=\sum _{i=1}^n{\tilde{H}}_{ni}.$$

To proceed with establishing the asymptotic normality of ${\widehat{\mathbb{Q}}}_n^{u,x}$, similar to Lemma 5.4 in [88], and applying the central limit theorem (CLT) for triangular arrays of martingale differences, for any fixed $x\in E$, $\{{\tilde{H}}_{ni},1\le i\le n\}$ (as found in [92], p. 23). In probability theory, the Cramér–Wold device is a fundamental result that provides the most effective approach to reducing the problem of convergence in distribution for random vectors ${\tilde{H}}_{ni}$, for $i=1,\dots ,n$ to that of convergence for all real-valued linear combinations $e^t{\tilde{H}}_{ni}$, for $i=1,\dots ,n$.

• Let $e\in {\mathbb{R}}^q$ such that $\parallel e\parallel \ne 0$, the $\{e^{\top }{\tilde{H}}_{ni},1\le i\le n\}$ forms a triangular array of stationary martingale differences with respect to the $\sigma$-field ${\mathcal{F}}_{i-1}$. In order to prove Lemma 12, we need to verify the following key conditions:

(a)

Conditional variance convergence.

$$\sum _{i=1}^n\mathbb{E}\left[{\left(e^{\top }{\tilde{H}}_{ni}\right)}^2\mid {\mathcal{F}}_{i-1}\right]\stackrel{\mathbb{P}}{\to }{\sigma }_u^2\left(x\right):=e^{\top }{\Sigma }^{\mathrm{x}}\left({\mathbb{Q}}^{u,x}\right)e,$$

(b)

Lindeberg condition [95].

$$n\mathbb{E}\left[{\left(e^{\top }{\tilde{H}}_{ni}\right)}^2{\mathbb{1}}_{\left|e^{\top }{\tilde{H}}_{ni}\right|>ϵ}\right]=o\left(1\right)$$

holds for any $ϵ>0$.

Proof of (a): Conditional variance convergence. Observe that

$$\left|\sum _{i=1}^n\mathbb{E}\left[{\left(e^{\top }{H}_{ni}\right)}^2\mid {\mathcal{F}}_{i-1}\right]-\sum _{i=1}^n\mathbb{E}\left[{\left(e^{\top }{\tilde{H}}_{ni}\right)}^2\mid {\mathcal{F}}_{i-1}\right]\right|\le \sum _{i=1}^n{\left(\mathbb{E}\left[e^{\top }{H}_{ni}\mid {\mathcal{F}}_{i-1}\right]\right)}^2.$$

(70)

Because ${\nabla }_m{\mathcal{G}}^{u,x}\left({\mathbb{Q}}_{u,x}\right)=0$ and by using a double conditioning with respect to ${\mathcal{A}}_{i-1}$ and conditions (A3)(i), (A4)(ii) and (A9), one may write

$$\begin{array}{c}{\displaystyle \left|\mathbb{E}\left(e^{\top }{H}_{ni}\mid {\mathcal{F}}_{i-1}\right)\right|}\hfill \\ =& \sqrt{{\displaystyle \frac{\varphi \left(h\right)}{n}}}{\displaystyle \frac{1}{\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}{e}^{\top }\left|\mathbb{E}\left[{\varrho }_i{\mathcal{K}}_i\left(x\right){\nabla }_m{\mathcal{G}}^{u,X_i}\left({\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right]\right|\mathrm{a}.\mathrm{s}.\hfill \\ \le & \sqrt{{\displaystyle \frac{\varphi \left(h\right)}{n}}}{\displaystyle \frac{1}{\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}{e}^{\top }\mathbb{E}\left\{{\varrho }_i{\mathcal{K}}_i\left(x\right)\left|{\nabla }_m{\mathcal{G}}^{u,X_i}\left({\mathbb{Q}}^{u,x}\right)-{\nabla }_m{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)\right|\mid {\mathcal{F}}_{i-1}\right\}\hfill \\ \le & \sqrt{{\displaystyle \frac{\varphi \left(h\right)}{n}}}\underset{v\in B(x,h)}{sup}\left|e^{\top }{\nabla }_m{\mathcal{G}}^{u,v}\left({\mathbb{Q}}^{u,x}\right)-e^{\top }{\nabla }_m{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)\right|{\displaystyle \frac{\mathbb{E}\left[{\varrho }_i{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]}{\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}\hfill \\ \le & \sqrt{{\displaystyle \frac{\varphi \left(h\right)}{n}}}\underset{x^{\prime }\in B(x,h)}{sup}\left|e^{\top }{\nabla }_m{\mathcal{G}}^{u,x^{\prime }}\left({\mathbb{Q}}^{u,x}\right)-e^{\top }{\nabla }_m{\mathcal{G}}^{u,x}\left({\mathbb{Q}}^{u,x}\right)\right|\hfill \\ & \times \left(\underset{x^{\prime }\in B(x,h)}{sup}\left|p\left(x^{\prime }\right)-p\left(x\right)\right|+p\left(x\right)\right){\displaystyle \frac{\mathbb{E}\left[{\mathcal{K}}_i\left(x\right)\mid {\mathcal{F}}_{i-1}\right]}{\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)}}.\hfill \end{array}$$

Making use of Lemma 2, one gets

$$\begin{array}{cc}\hfill \left|\mathbb{E}\left(e^{\top }{H}_{ni}\mid {\mathcal{F}}_{i-1}\right)\right|& =O\left(h^{\beta }\right)\sqrt{{\displaystyle \frac{\varphi \left(h\right)}{n}}}p\left(x\right)\left({\displaystyle \frac{f_{i,1}\left(x\right)}{f_1\left(x\right)}}+O_{a.s.}\left({\displaystyle \frac{g_{i,x}\left(h\right)}{\varphi \left(h\right)}}\right)\right).\hfill \end{array}$$

We conclude by A2(ii)–(iii) that

$$\sum _{i=1}^n{\left(\mathbb{E}\left(e^{\top }{H}_{ni}\mid {\mathcal{F}}_{i-1}\right)\right)}^2=O_{a.s}\left(h^{2\beta }\varphi \left(h\right)\right)=o_{a.s.}\left(1\right).$$

(71)

Therefore, the statement of (a) follows if we show that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\sum _{i=1}^n\mathbb{E}\left[{\left(e^{\top }{H}_{ni}\right)}^2\mid {\mathcal{F}}_{i-1}\right]\stackrel{\mathbb{P}}{\to }{e}^{\top }{\Sigma }^x\left({\mathbb{Q}}^{u,x}\right)e.\end{array}$$

(72)

To prove (72), use a double conditioning with respect to ${\mathcal{A}}_{i-1}$, one may write

$$\begin{array}{c}{\displaystyle \sum _{i=1}^n\mathbb{E}\left[{\left(e^{\top }{H}_{ni}\right)}^2\mid {\mathcal{F}}_{i-1}\right]}\hfill \\ =& {\displaystyle \frac{\varphi \left(h\right)/n}{{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^2}}\sum _{i=1}^n\mathbb{E}\left\{\mathbb{E}\left({\varrho }_i{\Delta }_i^2\left(x\right){\left(e^{\top }\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right)\right)}^2\mid {\mathcal{A}}_{i-1}\right)\mid {\mathcal{F}}_{i-1}\right\}\hfill \\ =& {\displaystyle \frac{\varphi \left(h\right)/n}{{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^2}}\sum _{i=1}^n\mathbb{E}\left\{{\varrho }_i{\Delta }_i^2\left(x\right){\Phi }_2^{X_i}\left({\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right\},\hfill \end{array}$$

(73)

where ${\Phi }_2^{X_i}\left({\mathbb{Q}}^{u,x}\right)$ is defined in (A8). Making use now the condition (A8), we get

$$\begin{array}{c}\hfill {\displaystyle \mathbb{E}\left\{{\varrho }_i{\Delta }_i^2\left(x\right){\Phi }_2^{X_i}\left({\mathbb{Q}}^{u,x}\right)\mid {\mathcal{F}}_{i-1}\right\}}\\ =& \mathbb{E}\left\{{\varrho }_i{\Delta }_i^2\left(x\right)\left({\Phi }_2^{X_i}\left({\mathbb{Q}}^{u,x}\right)-{\Phi }_2^x\left({\mathbb{Q}}^{u,x}\right)\right)\mid {\mathcal{F}}_{i-1}\right\}+{\Phi }_2^x\left({\mathbb{Q}}^{u,x}\right)\mathbb{E}\left\{{\varrho }_i{\Delta }_i^2\left(x\right)\mid {\mathcal{F}}_{i-1}\right\}\hfill \\ =& \mathbb{E}\left\{{\varrho }_i{\Delta }_i^2\left(x\right)\mid {\mathcal{F}}_{i-1}\right\}\left[o\left(\underset{x^{\prime }\in B(x,h)}{sup}\mid {\Phi }_2^{x^{\prime }}\left({\mathbb{Q}}^{u,x}\right)-{\Phi }_2^x\left({\mathbb{Q}}^{u,x}\right)\right)+{\Phi }_2^x\left({\mathbb{Q}}^{u,x}\right)\right]\hfill \\ =& \mathbb{E}\left\{p\left(X_i\right){\Delta }_i^2\left(x\right)\mid {\mathcal{F}}_{i-1}\right\}\left[{\Phi }_2^x\left({\mathbb{Q}}^{u,x}\right)+o\left(1\right)\right].\hfill \end{array}$$

By the application of Lemma 2, the statement (73) becomes

$$\begin{array}{ccc}\hfill \sum _{i=1}^n\mathbb{E}\left[{\left(e^{\top }{H}_{ni}\right)}^2\mid {\mathcal{F}}_{i-1}\right]& =& {\displaystyle \frac{1}{\frac{{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^2}{{\varphi }^2\left(h\right)}}}{\displaystyle \frac{1}{n\varphi \left(h\right)}}\sum _{i=1}^n\left\{{\mathcal{M}}_2\varphi \left(h\right)f_{i,1}\left(x\right)+O_{a.s.}\left(g_{i,x}\left(h\right)\right)\right\}\hfill \\ & \times & \left[{\Phi }_2^x\left({\mathbb{Q}}^{u,x}\right)+o\left(1\right)\right]\left[p\left(x\right)+o\left(1\right)\right]\hfill \\ & =& \left[{\Phi }_2^x\left({\mathbb{Q}}^{u,x}\right)p\left(x\right)+o\left(1\right)\right]{\displaystyle \frac{{\mathcal{M}}_2{f}_1\left(x\right)+o\left(1\right)}{{\mathcal{M}}_1^2{f}_1^2\left(x\right)+o\left(1\right)}}\hfill \\ & \underset{n\to \infty }{⟶}& {\displaystyle \frac{{\mathcal{M}}_2{\Phi }_2^x\left({\mathbb{Q}}^{u,x}\right)p\left(x\right)}{{\mathcal{M}}_1^2{f}_1\left(x\right)}}=:{\sigma }_u^2\left(x\right).\hfill \end{array}$$

(74)

Proof (b): Lindeberg condition. By properties of the conditional expectation and Jensen inequalities, it follows that

$$\begin{array}{ccc}\hfill {\tilde{H}}_{ni}^2& =& {\left(H_{ni}-\mathbb{E}\left[H_{ni}\mid {\mathcal{F}}_{i-1}\right]\right)}^2\hfill \\ & \le & 2H_{ni}^2+2{\left(\mathbb{E}\left[H_{ni}\mid {\mathcal{F}}_{i-1}\right]\right)}^2\hfill \\ & \le & 2H_{ni}^2+2\mathbb{E}\left[H_{ni}^2\mid {\mathcal{F}}_{i-1}\right],\hfill \end{array}$$

the fact that $|{\tilde{H}}_{ni}|\le 2|H_{ni}|$, which implies

$$\left\{\right|{\tilde{H}}_{ni}|>ϵ\}\subset \left\{\right|H_{ni}|>ϵ/2\}$$

and ${\tilde{H}}_{ni}^2\le 4H_{ni}^2$. We have

$${\tilde{H}}_{ni}^2{\mathbb{1}}_{\left\{\right|{\tilde{H}}_{ni}|>ϵ\}}\le 4H_{ni}^2{\mathbb{1}}_{\left\{\right|H_{ni}|>ϵ/2\}},$$

by taking the expectation and multiplying by n, we indeed obtain

$$n\mathbb{E}\left[{\tilde{H}}_{ni}^2{\mathbb{1}}_{\left\{\right|{\tilde{H}}_{ni}|>ϵ\}}\right]\le 4n\mathbb{E}\left[H_{ni}^2{\mathbb{1}}_{\left\{\right|H_{ni}|>ϵ/2\}}\right].$$

(75)

This enables us to control the tail of $H_{ni}$ and to analyze its behavior in the limit as its values grow large. Let $a>1$ and $b>1$ such that ${\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}=1$. Making use of Hölder and Markov inequalities, conditions (A8), and the first condition of (10) combined with Lemma 2, we get, for $ϵ>0$,

$$\mathbb{E}\left[H_{ni}^2{\mathbb{1}}_{\left\{\right|H_{ni}|>ϵ/2\}}\right]\le {\displaystyle \frac{\mathbb{E}|H_{ni}{|}^{2a}}{{\left(ϵ/2\right)}^{2a/b}}}.$$

Taking $a=1+\delta /2$, $C_1$ as positive constant, and by local continuous conditions (A8) and (A9), we obtain the following result:

$$\begin{array}{c}\hfill {\displaystyle 4n\mathbb{E}\left[H_{ni}^2{\mathbb{1}}_{\left\{\right|H_{ni}|>ϵ/2\}}\right]}\\ \le & C_1{\left({\displaystyle \frac{\varphi \left(h\right)}{n}}\right)}^{{\displaystyle \frac{2+\delta }{2}}}{\displaystyle \frac{n}{{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^{2+\delta }}}\mathbb{E}\left[|\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){\varrho }_i{\mathcal{K}}_i{\left(x\right)|}^{2+\delta }\right]\hfill \\ \le & C_1{\left({\displaystyle \frac{\varphi \left(h\right)}{n}}\right)}^{{\displaystyle \frac{2+\delta }{2}}}{\displaystyle \frac{n}{{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^{2+\delta }}}\mathbb{E}\left[\mathbb{E}\left(|\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){|}^{2+\delta }{\varrho }_i{\mathcal{K}}_i{\left(x\right)}^{2+\delta }\mid {\mathcal{A}}_{i-1}\right)\right]\hfill \\ \le & C_1{\left({\displaystyle \frac{\varphi \left(h\right)}{n}}\right)}^{{\displaystyle \frac{2+\delta }{2}}}{\displaystyle \frac{n}{{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^{2+\delta }}}\mathbb{E}\left[\mathbb{E}\left(|\mathbb{D}\left(u,Y_i-{\mathbb{Q}}^{u,x}\right){|}^{2+\delta }{\varrho }_i{\mathcal{K}}_i{\left(x\right)}^{2+\delta }\mid X_i\right)\right]\hfill \\ \le & C_1{\left({\displaystyle \frac{\varphi \left(h\right)}{n}}\right)}^{{\displaystyle \frac{2+\delta }{2}}}{\displaystyle \frac{n}{{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^{2+\delta }}}\mathbb{E}\left[{\Phi }_{2+\delta }^{X_i}\left({\mathbb{Q}}^{u,x}\right)p\left(X_i\right){\left({\mathcal{K}}_i\left(x\right)\right)}^{2+\delta }\right]\hfill \\ \le & C_1{\left({\displaystyle \frac{\varphi \left(h\right)}{n}}\right)}^{{\displaystyle \frac{2+\delta }{2}}}{\displaystyle \frac{n}{{\left(\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^{2+\delta }}}\mathbb{E}\left[{\left({\mathcal{K}}_i\left(x\right)\right)}^{2+\delta }\left({\Phi }_{2+\delta }^x\left({\mathbb{Q}}^{u,x}\right)+o\left(1\right)\right)\left(p\left(x\right)+o\left(1\right)\right)\right]\hfill \\ \le & C_1{\left(n\varphi \left(h\right)\right)}^{-\delta /2}{\displaystyle \frac{\frac{1}{\varphi \left(h\right)}\mathbb{E}\left[{\left({\mathcal{K}}_i\left(x\right)\right)}^{2+\delta }\right]}{{\left(\frac{1}{\varphi \left(h\right)}\mathbb{E}\left({\mathcal{K}}_1\left(x\right)\right)\right)}^{2+\delta }}}\left({\Phi }_{2+\delta }^x\left({\mathbb{Q}}^{u,x}\right)p\left(x\right)+o\left(1\right)\right)\hfill \\ \le & C_1{\left(n\varphi \left(h\right)\right)}^{-\delta /2}{\displaystyle \frac{{\mathcal{M}}_{2+\delta }{f}_1\left(x\right)+o\left(1\right)}{{\mathcal{M}}_1^{2+\delta }{f}_1^{2+\delta }\left(x\right)+o\left(1\right)}}\left({\Phi }_{2+\delta }^x\left({\mathbb{Q}}^{u,x}\right)p\left(x\right)+o\left(1\right)\right)\hfill \\ =& O\left({\left(n\varphi \left(h\right)\right)}^{-\delta /2}\right)=o\left(1\right).\hfill \end{array}$$

(76)

Thus, to achieve the proof of Theorem 3, it suffices to use Proposition 2, (13), Lemma 11, Lemma 12 and (70)–(76), and the Slulsky Theorem. □

Proof of Corollary 1.

Let us denote

$$Z_n^x={\displaystyle \frac{{\mathcal{M}}_{1,n}}{\sqrt{{\mathcal{M}}_{2,n}}}}\sqrt{{\displaystyle \frac{n{F}_{n,x}\left(h\right)p_n\left(x\right)}{{\Lambda }_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)}}}{\mathbb{M}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right).$$

Remark that

$$\begin{array}{ccc}\hfill Z_n^x& =& {\displaystyle \frac{{\mathcal{M}}_{1,n}\sqrt{{\mathcal{M}}_2}}{{\mathcal{M}}_1\sqrt{{\mathcal{M}}_{2,n}}}}\sqrt{{\displaystyle \frac{n{F}_{n,x}\left(h\right)p_n\left(x\right){\Lambda }^x\left({\mathbb{Q}}^{u,x}\right)}{p\left(x\right){\Lambda }_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)n\varphi \left(h\right)f_1\left(x\right)}}}{\mathbb{M}}_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right){\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]}^{-1}\hfill \\ & & \times {\displaystyle \frac{{\mathcal{M}}_1}{\sqrt{{\mathcal{M}}_2}}}\sqrt{{\displaystyle \frac{n\varphi \left(h\right)f_1\left(x\right)p\left(x\right)}{{\Lambda }^x\left({\mathbb{Q}}^{u,x}\right)}}}\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right)\hfill \\ & =& Z_{1,n}^x\times Z_{2,n}^x,\hfill \end{array}$$

(77)

by (14) and Theorem 1, we have $Z_{2,n}^x$ converges in distribution to a multivariate normal distribution with mean zero and covariance matrix equal to the identity covariance matrix ${\sigma }_0^2$, where

$$\begin{array}{ccc}\hfill {\sigma }_0^2& =& {\displaystyle \frac{{\mathcal{M}}_1^2}{{\mathcal{M}}_2}}n\varphi \left(h\right)f_1\left(x\right)p\left(x\right)\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]{\left({\Lambda }^x\left({\mathbb{Q}}^{u,x}\right)\right)}^{-1}\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]\mathbb{V}ar\left({\widehat{\mathbb{Q}}}_n^{u,x}-{\mathbb{Q}}^{u,x}\right)\hfill \\ & =& {\displaystyle \frac{{\mathcal{M}}_1^2}{{\mathcal{M}}_2}}{f}_1\left(x\right)p\left(x\right)\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]{\left({\Lambda }^x\left({\mathbb{Q}}^{u,x}\right)\right)}^{-1}\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]\hfill \\ & \times & {\displaystyle \frac{{\mathcal{M}}_2}{{\mathcal{M}}_1^2{f}_1\left(x\right)p\left(x\right)}}{\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]}^{-1}{\Lambda }^x\left({\mathbb{Q}}^{u,x}\right){\left[{\mathbb{M}}^x\left({\mathbb{Q}}^{u,x}\right)\right]}^{-1}\hfill \\ & =& {\mathbf{I}}_q.\hfill \end{array}$$

(78)

Now, to obtain the result of the corollary 1, it suffices to show that the first term $Z_{1,n}^x$ converges to 1 in probability. Similarly to the proof of Corollary 1 in [88], we have, as $n⟶\infty$,

$$\left\{\begin{array}{c}{\displaystyle \frac{F_{n,x}\left(h\right)}{\varphi \left(h\right)f_1\left(x\right)}}\stackrel{\mathbb{P}}{\to }1,\hfill \\ {\mathcal{M}}_{1,n}\stackrel{\mathbb{P}}{\to }{\mathcal{M}}_1,\hfill \\ {\mathcal{M}}_{2,n}\stackrel{\mathbb{P}}{\to }{\mathcal{M}}_2.\hfill \end{array}\right.$$

(79)

Furthermore, by Theorem 1 and Proposition 2, it follows that

$${\Lambda }_n^x\left({\widehat{\mathbb{Q}}}_n^{u,x}\right)\stackrel{\mathbb{P}}{\to }{\Lambda }^x\left({\mathbb{Q}}^{u,x}\right),\mathrm{as} n⟶\infty .$$

(80)

Finally, by Lemma 3, it follows that

$$p_n\left(x\right)\stackrel{\mathbb{P}}{\to }\mathbb{E}\left(\varrho \mid X=x\right)=\mathbb{P}\left(\varrho \mid X=x\right)=p\left(x\right),\mathrm{as} n⟶\infty .$$

(81)

Hence, $Z_{1,n}^x\stackrel{\mathbb{P}}{\to }1$ follows from (77) and (79)–(81). □