Optimal Minimax Rate of Smoothing Parameter in Distributed Nonparametric Specification Test

Abstract

A model specification test is a statistical procedure used to assess whether a given statistical model accurately represents the underlying data-generating process. The smoothing-based nonparametric specification test is widely used due to its efficiency against “singular” local alternatives. However, large modern datasets create various computational problems when implementing the nonparametric specification test. The divide-and-conquer algorithm is highly effective for handling large datasets, as it can break down a large dataset into more manageable datasets. By applying divide-and-conquer, the nonparametric specification test can handle the computational problems induced by the massive size of the modern datasets, leading to improved scalability and efficiency and reduced processing time. However, the selection of smoothing parameters for optimal power of the distributed algorithm is an important problem. The rate of the smoothing parameter that ensures rate optimality of the test in the context of testing the specification of a nonlinear parametric regression function is studied in the literature. In this paper, we verified the uniqueness of the rate of the smoothing parameter that ensures the rate optimality of divide-and-conquer-based tests. By employing a penalty method to select the smoothing parameter, we obtain a test with an asymptotic normal null distribution and adaptiveness properties. The performance of this test is further illustrated through numerical simulations.

1. Introduction

Big datasets characterized by large sample sizes N and/or high dimension p are increasingly accessible. In this paper, we focus on datasets with massive sample size N and low dimension p. However, directly making inferences from such large datasets is computationally infeasible due to limitations in processor memory, which makes selecting an appropriate model for big data particularly challenging. The divide-and-conquer approach is intuitive and has been widely employed across various fields to tackle diverse problems. Zaremba et al. [1] utilized this strategy to address two-sample test problems. In situations involving large sample sizes or high-dimensional predictors, Chen and Xie [2] applied the divide-and-conquer methodology for variable selection in generalized linear models. Battey et al. [3] integrated the divide-and-conquer algorithm with high-dimensional hypothesis testing and estimation. Additionally, as noted in [4], samples in big datasets are often aggregated from multiple sources. Therefore, feasible and robust specification testing methods, essential for addressing model misspecification, are critical for handling massive datasets.

Suppose we have a sequence of independent observations ${\left\{(y_i,{\mathbf{x}}_i)\right\}}_{i=1}^N$ drawn from a population $(Y,\mathbf{X})\in \mathbb{R}\times {[0,1]}^p$, where the unknown regression function $E(Y\mid \mathbf{X}=\mathbf{x})=m\left(\mathbf{x}\right)$ is assumed to be smooth. In this context, a specification test is necessary to assess the functional form of the regression and justify the use of a parametric model. Given a parametric family of known real functions $g(\mathbf{x};\mathit{\theta })$, the null and alternative hypotheses can be described as follows:

$$\begin{array}{cc}\hfill & H_0:m\left(\mathbf{x}\right)=g(\mathbf{x},{\mathit{\theta }}_0)\mathrm{for} \mathrm{some}{\mathit{\theta }}_0\in \Theta ,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & H_1:m\left(\mathbf{x}\right)\ne g(\mathbf{x},{\mathit{\theta }}_0)\mathrm{for} \mathrm{all}{\mathit{\theta }}_0\in \Theta ,\hfill \end{array}$$

(2)

where $\Theta \subset {\mathbb{R}}^q$ denotes the parameter space. This hypothesis testing problem has been widely studied in the literature. One category of approach is to measure the distance between the estimator under the null and the nonparametric estimator under alternative models (see Hardle and Mammen [5], Neumeyer and Van Keilegom [6], González-Manteiga and Crujeiras [7] and the references therein). Another competing approach relies on the empirical process of the residuals from the parametric model [8,9]. An important criterion for evaluating the behavior of these tests is their power performance under local alternatives (see, e.g., [10]). Additionally, Ingster [11,12] proposed an alternative approach to investigating the asymptotic power properties of tests via the minimax approach. Guerre and Lavergne [13] further provided the optimal minimax rate for the smoothing parameter that ensures the rate optimality of the test in the context of testing the specification of a nonlinear parametric regression function. Conditional on a subset of covariates in regression modeling, Cai et al. [14] proposed a significance test for the partial mean independence problem based on machine learning methods and data splitting. Tan and Zhu [15] proposed a residual-marked empirical process that adapts to the underlying model, forming the basis of a goodness-of-fit test for parametric single-index models with a diverging number of predictors. However, existing methods that work well for moderate-sized datasets are not feasible for massive datasets due to computational limitations. Han et al. [16] developed an optimal sampling strategy to select a small subset from a large pool of data to reduce the computation budget for model checking big data. When dealing with test statistics that are quadratic forms [5,17], the computational complexity of the quadratic form test statistic is $O\left(N^2\right)$, which presents a significant computational burden for large-scale data.

To address this issue, a divide-and-conquer-based test statistic was proposed in [18,19]. Zhao et al. [18] incorporated a divide-and-conquer strategy into [17] a nonparametric test statistic along with a data-driven bandwidth selection procedure. However, this integrated approach can easily inflate the type I error rate. To mitigate issues associated with choosing smoothing parameters and preserving the type-I error rate, Ref. Zhao et al. [19] proposed randomly splitting the observations into two subsets. In the first subset, an “optimal” smoothing parameter is selected based on a straightforward criterion. Subsequently, a lack-of-fit test grounded in asymptotic theory is conducted using the second subset. This data-splitting strategy effectively controls the type-I error rate. However, the sample splitting will reduce the power, as only a subset of the sample is used to construct the test statistics. Furthermore, the uniqueness of the rate of the smoothing parameter, which ensures the rate optimality of the divide-and-conquer-based test statistic, is not addressed in [18,19]. In this paper, we establish and verify the uniqueness of the rate for the smoothing parameter that guarantees the rate optimality of the divide-and-conquer-based test statistic.

Moreover, it is well known that the optimal smoothing parameters for testing differ from those that are optimal for estimation [11,12,20]. As a result, there has been growing interest in adaptive testing methods. One approach is to consider a set of suitable values for the bandwidth and proceed from there, as discussed in [21,22]. In this paper, we integrate the smoothing parameter selection method in [22] with the divide-and-conquer-based test statistic proposed in [18]. This combination leads to a computationally feasible and adaptive test statistic which retains its asymptotic normality under the null hypothesis.

The paper is organized as follows: Section 2 describes the test statistics and their corresponding asymptotic behavior under the null hypothesis. In Section 3, we demonstrate the unique rate of the smoothing parameter that ensures rate optimality in the DZH test. Section 4 presents simulation studies for illustration. The proofs of the theorems are provided in Section 5.

2. The Divide-And-Conquer-Based Test Statistics

The distributed test statistic proposed in Zhao et al. [18] is based on the test statistic in Zheng [17], where the kernel method is used to estimate the conditional moment $E\left\{{\zeta }_{i}E({\zeta }_i\mid {\mathbf{X}}_i)f\left({\mathbf{X}}_i\right)\right\}$, ${\zeta }_i=y_i-g({\mathbf{x}}_i;{\mathit{\theta }}_0)$, and $f(·)$ is the density function of ${\mathbf{x}}_i$. The kernel-based sample estimator of the quantity is

$$Q_N\left(h_N\right)={\displaystyle \frac{1}{N(N-1)}}\sum _{i=1}^N\sum _{j\ne i}^N{\mathcal{K}}_{h_N}\left({\mathbf{x}}_i-{\mathbf{x}}_j\right)e_i{e}_j,$$

where ${\mathcal{K}}_{h_N}(·)=\mathcal{K}(·/h_N)/h_N^p$ denotes a p-dimensional kernel function, $h_N$ is the bandwidth depending on N, $e_i=y_i-g({\mathbf{x}}_i,{\widehat{\mathit{\theta }}}_N)$, and ${\widehat{\mathit{\theta }}}_N$ is an estimate of ${\mathit{\theta }}_0$ under the null hypothesis.

When handling exceptionally large datasets where the sample size N becomes unmanageable, the test statistics combined with the divide-and-conquer procedure is proposed in Zhao et al. [18]. Initially, the dataset is partitioned into K equally sized subsets, each containing n observations. The test statistic based on the observations in the kth subset is

$$V_k\left(h_n\right)={\displaystyle \frac{1}{n(n-1)}}\sum _{i=1}^n\sum _{j\ne i}^n{\mathcal{K}}_{h_n}\left({\mathbf{x}}_{ik}-{\mathbf{x}}_{jk}\right)e_{ik}{e}_{jk},$$

where $e_{ik}$’s are the fitted residuals with ${\left\{({\mathbf{x}}_{ik},y_{ik})\right\}}_{i=1}^n$. As $n{h}_n^{p/2}{V}_k\left(h_n\right)$ is asymptotically normal with mean zero and ${\delta }^2$ under mild conditions [17], where

$${\delta }^2=2\int {\mathcal{K}}^2\left(\mathbf{u}\right)d\mathbf{u}\int {\left\{{\sigma }^2\left(\mathbf{x}\right)\right\}}^2{f}^2\left(\mathbf{x}\right)d\mathbf{x},{\sigma }^2\left(\mathbf{x}\right)=E({\epsilon }_i^2\mid \mathbf{x}),{\epsilon }_i=y_i-m\left({\mathbf{x}}_i\right)$$

Then, Zhao et al. [18] combined the test statistic by taking an average,

$$\begin{array}{c}\hfill T_N\left(h_n\right)={\displaystyle \frac{1}{K}}\sum _{k=1}^K{\displaystyle \frac{V_k\left(h_n\right)}{\widehat{\delta }\left(h_n\right)}},\end{array}$$

(3)

where ${\widehat{\delta }}^2\left(h_n\right)$ is an estimate of ${\delta }^2$. A natural estimator is ${\widehat{\delta }}^2\left(h_n\right)=K^{-1}{\sum }_{k=1}^K{\widehat{\delta }}_k^2\left(h_n\right)$, where

$${\widehat{\delta }}_k^2\left(h_n\right)={\displaystyle \frac{2}{n(n-1)}}\sum _{i=1}^n\sum _{j\ne i}^n{h}_n^p{\mathcal{K}}_{h_n}^2\left({\mathbf{x}}_{ik}-{\mathbf{x}}_{jk}\right)e_{ik}^2{e}_{jk}^2$$

is a consistent estimate of ${\delta }^2$ based on the kth subset. The test based on statistic $T_N\left(h_n\right)$ is denoted as the DZH test in [18]. $T_N\left(h_n\right)$ is asymptotic normal under the null hypothesis provided some mild conditions [18]. In this paper, we study the asymptotic behavior of $T_N\left(h_n\right)$ by relaxing $n{h}_n^p/lnn\to \infty$ to $n{h}_n^p\to \infty$.

Assumption 1.

The density function $f\left(\mathbf{x}\right)$ of $\mathbf{x}$ and its first-order derivatives are uniformly bounded, $0\le \underline{f}\le f\left(\mathbf{x}\right)\le \overline{f}<\infty$, $\forall \mathbf{x}\in {[0,1]}^p$.

Assumption 2.

Suppose that $E\left({\epsilon }_i\right|{\mathbf{X}}_i)=0$ and ${\sigma }^2\left({\mathbf{x}}_i\right)\le {\overline{\sigma }}^2$, $E\left({\epsilon }_i^4\right|{\mathbf{X}}_i)={\sigma }^4\left({\mathbf{x}}_i\right)\le C$ uniformly in i. We also assume that ${\sigma }^2\left({\mathbf{x}}_i\right)$ is differentiable and that its first-order derivatives are uniformly bounded for all i.

Assumption 3.

For any $m(·)$, not necessarily in $H_0$, let

$${\mathit{\theta }}^{*}=\underset{\mathit{\theta }\in \Theta }{argmin}E{\{m\left(\mathbf{X}\right)-g(\mathbf{X};\mathit{\theta })\}}^2.$$

Under $H_0$, ${\mathit{\theta }}^{*}={\mathit{\theta }}_0$. For any $m(·)$, ${\mathit{\theta }}^{*}$ is unique. ${\widehat{\mathit{\theta }}}_n$ is the estimator of ${\mathit{\theta }}^{*}$ such that $\sqrt{nK}({\widehat{\mathit{\theta }}}_n-{\mathit{\theta }}^{*})=O_p\left(1\right)$ uniformly with respect to $m(·)$ with $E\left\{m^4\left(\mathbf{X}\right)\right\}\le C<\infty$, i.e.,

$$\forall \eta >0,\exists ϵ>0:\underset{n,K\to \infty }{lim\; sup}\underset{E\left\{m^4\left(X\right)\right\}\le C}{sup}P\left(\sqrt{nK}\parallel {\widehat{\mathit{\theta }}}_n-{\mathit{\theta }}^{*}\parallel >ϵ\right)\le \eta .$$

Assumption 4.

$g(·,·)$ is uniformly bounded in $\mathbf{x}$ and θ, is twice continuously differentiable with respect to θ, with first- and second-order derivatives $g_{\theta }(·,·)$ and $g_{\theta \theta }(·,·)$ uniformly bounded in $\mathbf{x}$ and $\mathit{\theta }\in \Theta$ with upper bound ${\overline{g}}_{\theta }$ and ${\overline{g}}_{\theta \theta }$, respectively.

Assumption 5.

$\mathcal{K}\left(u\right)$ is a nonnegative, bounded, continuous, and symmetric function such that $\int \mathcal{K}\left(u\right)du=1$.

Assumption 6.

Suppose $\mathcal{K}\left(u\right)$’s Fourier transform $\widehat{\mathcal{K}}\left(u\right)=\int exp(-i{t}^{\prime }u)\mathcal{K}\left(t\right)dt$ is strictly positive on its nonempty support.

Theorem 1

(Null hypothesis). Suppose Assumptions 1–5 hold; if $n{h}_n^p\to \infty$, $h_n\to 0$ and $K\to \infty$, then we have $n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)\stackrel{d}{\to }N(0,1)$.

This result suggests that we can reject $H_0$ at an $\alpha$ level of significance if the normalized $n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)$ is larger than $z_{\alpha }$, where $z_{\alpha }$ is the upper $\alpha$th quantile of the standard normal distribution. Given that our focus is to demonstrate the null asymptotic results under the specific bandwidth $h_n$ in Theorem 1, the condition can be relaxed to $n{h}_n^p\to \infty$. The proof of this theorem closely resembles that of Theorem 1 in Zhao et al. [18], with the exception that we relax the condition from $n{h}_n^p/lnn\to \infty$ to $n{h}_n^p\to \infty$. Therefore, we omit the detailed proof here. However, how to choose an appropriate K via balancing the computation budget and statistical efficiency in practical applications is still an open question.

To develop an adaptive test, we integrate the smoothing parameter selection procedure proposed by Guerre and Lavergne [22] and $T_N\left(h_n\right)$. This procedure advocates for a larger smoothing parameter under the null hypothesis and selects h based on this criterion.

$$h_n^{*}=\underset{h\in {\mathcal{H}}_n}{\mathrm{argmax}}\{K^{-1}\sum _{k=1}^K{V}_k\left(h\right)-{\gamma }_n{\widehat{\upsilon }}_{h,h_0}\}.$$

where ${\mathcal{H}}_n$ is the given candidate set of $h_n$ and ${\widehat{\upsilon }}_{h,h_0}$ is an estimator of the asymptotic null standard deviation of $K^{-1}{\sum }_{k=1}^K\left(V_k\left(h\right)-V_k\left(h_0\right)\right)$. The asymptotic null standard deviation of $K^{-1}{\sum }_{k=1}^K(V_k\left(h\right)-V_k\left(h_0\right))$ is

$${\upsilon }_{h,h_0}=\sqrt{{\displaystyle \frac{2}{Kn(n-1)}}\int {\left[{\mathcal{K}}_h(x_1-x_2)-{\mathcal{K}}_{h_0}(x_1-x_2)\right]}^2{\sigma }^2\left(x_1\right){\sigma }^2\left(x_2\right)f\left(x_1\right)f\left(x_2\right)d{x}_{1}d{x}_2},$$

where an intuitive estimator is

$${\widehat{\upsilon }}_{h,h_0}=\sqrt{{\displaystyle \frac{2}{K^2{n}^2{(n-1)}^2}}\sum _{k=1}^K\sum _{i=1}^n\sum _{i\ne j}{\left[{\mathcal{K}}_h(x_{ik}-x_{jk})-{\mathcal{K}}_{h_0}(x_{ik}-x_{jk})\right]}^2{e}_{ik}^2{e}_{jk}^2}.$$

Let $h_0$ represent the largest element in ${\mathcal{H}}_n$. The corresponding test statistic based on the selected $h_n^{*}$ is given by

$$T_N^{*}\left(h_n^{*}\right)={\displaystyle \frac{1}{K}}\sum _{k=1}^K{\displaystyle \frac{V_k\left(h_n^{*}\right)}{\widehat{\delta }\left(h_0\right)}}$$

Under the null hypothesis, as ${\gamma }_n\to \infty$, the test statistic $T_N^{*}\left(h_n^{*}\right)$ tends to favor $T_N\left(h_0\right)$. Given that $T_N\left(h_0\right)$ is asymptotically normal under Assumptions 1–5, and considering that $n{h}_0^p\to \infty$, $h_0\to 0$ and $K\to \infty$, $T_N^{*}\left(h_n^{*}\right)$ also achieves asymptotic normality under the additional condition that ${\gamma }_n\to \infty$. Moreover, this statistic exhibits an adaptive property, enhancing its suitability across a broader range of alternative hypotheses.

3. The Unique Rate of the Smoothing Parameter Ensuring Rate-Optimality in the DZH Test

Our previous study in Zhao et al. [18] demonstrated that the optimal power of the test is significantly dependent on the set of bandwidth candidates, ${\mathcal{H}}_n$. This set should encompass the optimal rate of bandwidth, ${\tilde{h}}_n\asymp N^{-2/(4s+p)}$ to achieve desired performance. However, the uniqueness of this bandwidth rate was not established. In this section, we will demonstrate the uniqueness of the rate $O\left(N^{-2/(4s+p)}\right)$, which is critical for ensuring the rate-optimality of the DZH test. Let the Hölder class $C_p(L,s)$ be the set of maps ${\mathsf{\ell }}_N(·)$ from ${[0,1]}^p$ to $\mathbb{R}$ with

$$\begin{array}{cc}\hfill C_p(L,s)& =\left\{{\mathsf{\ell }}_N(·): |{\mathsf{\ell }}_N\left(\mathbf{x}\right)-{\mathsf{\ell }}_N{\left(\mathbf{y}\right)|\le L\parallel \mathbf{x}-\mathbf{y}\parallel }^s\mathrm{for} \mathrm{all}\mathbf{x},\mathbf{y}\mathrm{in}{[0,1]}^p\right\},s\in (0,1],\hfill \\ \hfill C_p(L,s)& =\left\{{\mathsf{\ell }}_N(·): \mathrm{the}\lfloor s\rfloor \mathrm{th} \mathrm{partial} \mathrm{derivatives} \mathrm{of}{\mathsf{\ell }}_N(·)\mathrm{are} \mathrm{in}{C}_p(L,s-\lfloor s\rfloor )\right\},s>1,\hfill \end{array}$$

where $\lfloor s\rfloor$ is the lower integer part of s. Consider the following alternative hypothesis:

$$H_1\left({\kappa }_N{\rho }_N\right)=\{m_N\left(\mathbf{x}\right):E{\mathsf{\ell }}_N^2(\mathbf{X},{\mathit{\theta }}^{*})\ge {\kappa }_N^2{\rho }_N^2,{\mathsf{\ell }}_N(·)\in C_p(L,s)\mathrm{for}\mathrm{fixed}s>p/4\},$$

where ${\mathsf{\ell }}_N(·)=m_N(·)-g(·;{\mathit{\theta }}^{*})$ and ${\rho }_N=N^{-2s/(4s+p)}$. ${\rho }_N$ is the optimal minimax rate for nonparametric specification testing in regression models of known $s>p/4$ for the Hölder class given above (see Guerre and Lavergne [13]).

Theorem 2.

Suppose Assumptions 1–6 hold, ${\tilde{h}}_n\asymp N^{-2/(4s+p)}$ is the only bandwidth rate such that $I\{n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)\ge z_{\alpha }\}$ can consistent uniformly against ${\left\{m_N(·)\right\}}_{N\ge 1}\in H_1\left({\kappa }_N{\rho }_N\right)$ for any ${\kappa }_N\to \infty$.

4. Simulation Studies

In this section, we present simulation studies to examine the behaviors of the size and power of the tests based on test statistics $T_N\left(h_n\right)$ and $T^{*}\left(h_n^{*}\right)$, denoted as DZH and MD, respectively. We choose $p=2,N=2000,4000,8000,K=10,20,40$. To demonstrate the adaptiveness of the MD test compared to the DZH test and to maintain the type I error rate for both tests, we select ${\mathcal{H}}_n=\{h_1=0.28,h_2=0.14,h_3=0.07\}$. For the MD test, we adopt a penalty sequence ${\gamma }_n=c\sqrt{2ln\left(\right|{\mathcal{H}}_n\left|\right)}$, where $c=2$, as recommended in Guerre and Lavergne [22], and $|{\mathcal{H}}_n|$ represents the cardinality of ${\mathcal{H}}_n$. Two models are used to generate response variable Y.

• M1: $Y=1+X_1+X_2+\epsilon$
• M2: $Y=1+X_1+X_2+sin\left(b{X}_1\right)+\epsilon ,b\in \{0.8,1,10\}$

We define the variables $X_1$ and $X_2$ where $X_1=Z_1$ represents a simple assignment, and $X_2=Z_1+Z_2/\sqrt{2}$ combines the influences of two independent factors under transformation to maintain variance consistency. To rigorously test the robustness of our proposed statistical method against different models, $Z_1$ and $Z_2$ are independently drawn from either the standard normal distribution, which provides a baseline due to its well-known properties, or from the Student’s t-distribution with 5 degrees of freedom, known for its heavier tails and greater kurtosis. This choice enables an examination of the test’s sensitivity deviating from normality. Furthermore, to assess the impact of error distributions on test performance, we explore three different distributions for the error term $\epsilon$. These include the standard normal distribution, standardized exponential distribution, and Student’s t-distribution with 5 degrees of freedom. Standard normal distribution assumes ideal conditions. The standardized exponential distribution introduces asymmetry and is skewed. Student’s t-distribution with 5 degrees of freedom tests the resilience of the method against errors with heavier tails. This comprehensive approach allows us to determine the test’s effectiveness and reliability across various scenarios reflective of real-world data complexities. The kernel function $\mathcal{K}$ used is the bivariate standard normal density function. M1 is used to assess the size of the tests. To investigate the power of the test against a high(low)-frequency alternatives, M2 is considered. In model M2, small(large) values of b represent low(high)-frequency alternatives. b is selected to be $\{0.8,1,10\}$.

The empirical sizes are reported in

Table 1. Empirical sizes for different values of N with $\alpha =1\%,5\%,10\%$ when the error term is normal distribution.

		MD			DZH
				${\mathit{h}}_{\mathbf{1}}$			${\mathit{h}}_{\mathbf{2}}$			${\mathit{h}}_{\mathbf{3}}$
$\mathit{N}$	$\mathit{\alpha }$∖$\mathit{K}$	10	20	40	10	20	40	10	20	40	10	20	40
	1%	1.3	1.3	1.1	0.8	1.0	1.0	0.7	0.7	0.6	0.6	0.9	0.8
2000	5%	5.1	5.8	5.3	4.4	5.3	4.8	5.3	4.3	4.4	4.6	5.8	5.5
	10%	11.1	10.1	11.3	9.9	9.7	10.0	10.3	10.0	10.7	11.6	12.1	11.6
	1%	1.3	1.2	1.1	1.0	1.0	0.9	1.0	0.8	0.7	0.9	0.5	0.6
4000	5%	5.4	5.7	4.8	4.7	5.1	4.2	4.9	4.6	4.6	5.6	4.9	4.3
	10%	9.7	10.4	9.8	9.2	9.9	8.9	9.2	8.8	9.8	10.1	9.2	8.8
	1%	0.5	1.0	0.5	0.4	0.7	0.4	0.4	0.8	0.5	1.0	1.0	0.6
8000	5%	5.2	5.1	4.1	4.2	4.5	3.4	5.0	4.9	4.3	5.2	5.5	4.3
	10%	10.8	10.7	10.2	10.1	9.5	9.3	9.9	9.4	9.3	10.9	10.4	8.7

,

Table 2. Empirical sizes for different values of N with $\alpha =1\%,5\%,10\%$ when the error term is exponential distribution.

		MD			DZH
					${\mathit{h}}_{\mathbf{1}}$			${\mathit{h}}_{\mathbf{2}}$			${\mathit{h}}_{\mathbf{3}}$
$\mathit{N}$	$\mathit{\alpha }$∖$\mathit{K}$	10	20	40	10	20	40	10	20	40	10	20	40
	1%	1.7	2.2	1.6	1.5	1.9	1.5	0.7	0.8	1.5	1.1	0.7	1.1
2000	5%	5.7	5.5	5.5	5.2	5.1	5.0	5.8	4.9	4.9	6.2	5.1	5.1
	10%	10.6	10.8	9.9	10.0	9.9	9.6	12.1	10.5	9.7	11.5	11.0	9.8
	1%	1.3	0.8	0.7	1.2	0.5	0.7	1.0	0.5	1.2	1.2	0.6	0.6
4000	5%	5.8	5.2	4.8	4.9	4.6	4.3	5.1	4.9	4.6	4.8	4.9	3.8
	10%	11.3	9.7	10.4	10.3	9.0	9.8	8.9	9.4	9.3	8.9	9.8	8.6
	1%	1.0	1.1	1.0	0.9	1.0	0.8	1.4	1.2	0.5	1.1	1.3	0.6
8000	5%	4.9	5.5	3.4	4.3	4.5	2.9	4.5	4.6	2.9	4.7	5.4	4.1
	10%	10.1	10.3	8.1	9.3	9.5	7.4	8.3	9.4	8.3	9.0	9.3	9.8

,

Table 3. Empirical sizes for different values of N with $\alpha =1\%,5\%,10\%$ when the error term is Student’s t distribution.

		MD			DZH
					${\mathit{h}}_{\mathbf{1}}$			${\mathit{h}}_{\mathbf{2}}$			${\mathit{h}}_{\mathbf{3}}$
$\mathit{N}$	$\mathit{\alpha }$∖$\mathit{K}$	10	20	40	10	20	40	10	20	40	10	20	40
	1%	1.1	0.7	1.1	1.1	0.6	0.7	0.6	0.6	0.4	0.7	0.6	0.6
2000	5%	4.8	4.7	4.7	4.3	4.2	4.4	4.0	4.0	4.9	4.3	3.8	4.8
	10%	9.9	9.8	10.4	9.2	9.1	9.2	8.0	8.6	10.1	10.4	8.8	9.0
	1%	1.2	1.6	1.5	1.1	1.1	1.2	1.0	1.3	1.1	0.8	1.1	0.6
4000	5%	5.3	5.6	5.8	4.9	5.1	5.2	5.0	5.5	5.5	5.2	5.4	5.2
	10%	10.0	10.4	10.5	9.0	9.5	9.5	10.2	10.4	9.1	11.6	9.3	9.0
	1%	1.2	1.0	1.5	1.0	0.9	1.2	0.5	0.5	1.3	0.6	0.9	1.1
8000	5%	5.2	3.7	5.2	4.5	3.2	4.5	3.4	3.4	4.8	4.7	4.4	5.0
	10%	9.1	8.8	9.9	8.3	7.9	9.1	8.5	8.6	10.1	9.9	10.1	11.4

and

Table 4. Empirical sizes for different values of N with $\alpha =1\%,5\%,10\%$ when the $Z_1$ and $Z_2$ are generated from Student’s t(5) distribution.

	MD				DZH
				${\mathit{h}}_{\mathbf{1}}$			${\mathit{h}}_{\mathbf{2}}$			${\mathit{h}}_{\mathbf{3}}$
$\mathit{N}$	$\mathit{\alpha }$∖$\mathit{K}$	10	20	40	10	20	40	10	20	40	10	20	40
	1%	1.2	1.3	1.7	0.9	1.1	1.6	1.5	0.8	1.1	1.0	1.1	1.1
2000	5%	6.0	6.6	6.0	5.6	6.1	5.3	5.0	5.8	5.9	5.3	5.9	5.7
	10%	10.9	10.8	11.5	10.0	10.3	10.5	10.2	10.4	10.2	10.9	10.9	11.0
	1%	0.9	0.5	1.1	0.7	0.5	0.8	0.9	1.0	1.4	1.2	1.3	1.1
4000	5%	5.7	4.9	4.9	4.9	4.3	4.6	4.8	4.2	4.5	5.4	5.4	5.3
	10%	11.3	10.4	9.9	9.6	9.8	8.6	10.1	9.2	9.6	10.8	9.8	10.1
	1%	0.9	1.8	0.5	0.8	1.2	0.3	0.7	1.1	1.1	0.8	1.0	0.6
8000	5%	4.6	5.6	4.6	4.0	4.7	4.3	3.9	5.9	4.3	4.3	4.2	5.5
	10%	9.8	11.7	9.4	9.1	11.1	8.8	9.1	10.8	8.7	8.8	9.4	10.5

for different error distributions, demonstrating that both methods effectively maintain the Type-I error rate. We describe the variation trends of the power along sample size N and K in

Table 5. Empirical power of MD and DZH with $\alpha =5\%$ when the error term is normal distribution.

	MD				DZH
				${\mathit{h}}_{\mathbf{1}}$			${\mathit{h}}_{\mathbf{2}}$			${\mathit{h}}_{\mathbf{3}}$
$\mathit{N}$	b∖$\mathit{K}$	10	20	40	10	20	40	10	20	40	10	20	40
2000	1	83.4	60.9	36.8	81.9	59	35.5	45.1	25.9	14.5	17.3	10.9	7.4
	10	100	100	100	18.2	12.4	8.4	100	100	100	100	100	100
4000	1	100	99	88.9	100	98.9	88.3	94.3	74.4	46.5	48.5	27	15.7
	10	100	100	100	62.8	36.6	20	100	100	100	100	100	100
8000	1	100	100	100	100	100	100	100	99.9	94.9	96.3	75.9	48.4
	10	100	100	100	100	93.5	67.1	100	100	100	100	100	100

and

Table 6. Empirical power of MD and DZH with $\alpha =5\%$ when the error term is exponential distribution.

	MD			DZH
				${\mathit{h}}_{\mathbf{1}}$			${\mathit{h}}_{\mathbf{2}}$			${\mathit{h}}_{\mathbf{3}}$
$\mathit{N}$	b∖$\mathit{K}$	10	20	40	10	20	40	10	20	40	10	20	40
2000	1	82.7	61.2	38.2	82.4	59.9	36.7	44.4	26.5	16.1	15.8	9.6	5.6
	10	100	100	100	19.1	11.8	7.5	100	100	99.9	100	100	100
4000	1	99.8	98.5	89.7	99.8	98.4	88.9	94.9	72	44.1	46.9	25.2	13.3
	10	100	100	100	66.3	34.2	19.1	100	100	100	100	100	100
8000	1	100	100	100	100	100	100	100	99.7	96.1	97	78.8	50.6
	10	100	100	100	99.8	94.1	66.4	100	100	100	100	100	100

and Figure 1, Figure 2, Figure 3 and Figure 4. DZH tests under three distinct bandwidths are included in the power comparison tables. The results indicate that power loss increases with larger K, a consequence of the information loss inherent in the divide-and-conquer procedure. For low-frequency alternatives (when $b=0.8,1$), the power of the DZH test improves as h increases. However, for the high-frequency setting $b=10$, DZH test exhibits an opposite trend with changes in h. The MD test performs comparably to the best scenarios of the DZH test for both low- and high-frequency alternatives, demonstrating its adaptive capability. This adaptiveness makes it suitable for a broader range of alternative hypotheses, accommodating both low- and high-frequency variations. The comparison between Figure 1 and Figure 4 demonstrates that the proposed test exhibits higher power when the variables $Z_1$ and $Z_2$ are generated from the Student’s t(5) distribution rather than the normal distribution. This underscores the robustness of our method against the heavy-tailed distribution of variables. The power performance of all the tests under the exponential distribution is comparable to that under the normal distribution. However, there is a noticeable decrease in power when the underlying model is the Student’s t distribution compared to the other two scenarios. All analyses were conducted using R version 4.3.2.

5. Proofs

Some Lemmas

In this section, we restate three Lemmas in Zhao et al. [18], omitting detailed proofs for brevity. Lemma 2 is restated under assumption $n{h}_n^p\to \infty$. We assume $q=1$ without loss of generality. Denote

$$e_{ik}=y_{ik}-g({\mathbf{x}}_{ik};{\widehat{\theta }}_n),{\epsilon }_{ik}=y_{ik}-m\left({\mathbf{x}}_{ik}\right),u_{ik}=g({\mathbf{x}}_{ik};{\widehat{\theta }}_n)-g({\mathbf{x}}_{ik};{\theta }^{*}),{\mathsf{\ell }}_{ik}=m\left({\mathbf{x}}_{ik}\right)-g({\mathbf{x}}_{ik};{\theta }^{*})$$

We introduce the following matrix notations:

$${\mathbf{e}}_k={\left(e_{ik},1\le i\le n\right)}^{\prime },{\epsilon }_k={\left({\epsilon }_{ik},1\le i\le n\right)}^{\prime },{\mathbf{u}}_k={\left(u_{ik},1\le i\le n\right)}^{\prime },{\mathsf{\ell }}_k={\left({\mathsf{\ell }}_{ik},1\le i\le n\right)}^{\prime }$$

$$W_k\left(h_n\right)={\left[w_{ik,jk}\right]}_{1\le i,j\le n}$$

$$w_{ik,jk}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{n(n-1)}}{\mathcal{K}}_{h_n}\left({\mathbf{x}}_{ik}-{\mathbf{x}}_{jk}\right),\hfill & i\ne j;\hfill \\ 0,\hfill & i=j\hfill \end{array}\right.$$

Under $H_1$, ${\mathsf{\ell }}_{ik}\ne 0$,

$$\begin{array}{cc}\hfill V_k& ={({\mathsf{\ell }}_k+{\epsilon }_k)}^{\prime }{W}_k\left(h_n\right)({\mathsf{\ell }}_k+{\epsilon }_k)-2{\mathsf{\ell }}_k^{\prime }{W}_k\left(h_n\right){\mathbf{u}}_k-2{\epsilon }_k^{\prime }{W}_k\left(h_n\right){\mathbf{u}}_k+{\mathbf{u}}_k^{\prime }{W}_k\left(h_n\right){\mathbf{u}}_k\hfill \\ \hfill & =V_{1k}-2V_{2k}-2V_{3k}+V_{4k}\hfill \end{array}$$

Under $H_0$, ${\mathsf{\ell }}_{ik}=0$, then $V_k=V_{1k}-2V_{3k}+V_{4k}$, where $V_{1k}={\epsilon }_k^{\prime }{W}_k\left(h_n\right){\epsilon }_k$. Following Zhao et al. [18], we decompose $n{h}_n^{p/2}{K}^{1/2}{T}_N$ in the following way:

$$\begin{array}{cc}\hfill n{h}_n^{p/2}{K}^{1/2}{T}_N& =n{h}_n^{p/2}{\widehat{\delta }}^{-1}{K}^{-1/2}\sum _{k=1}^K{V}_k\hfill \\ \hfill & =n{h}_n^{p/2}{\widehat{\delta }}^{-1}{K}^{-1/2}\sum _{k=1}^K{V}_{1k}-2n{h}_n^{p/2}{\widehat{\delta }}^{-1}{K}^{-1/2}\sum _{k=1}^K{V}_{2k}\hfill \\ \hfill & -2n{h}_n^{p/2}{\widehat{\delta }}^{-1}{K}^{-1/2}\sum _{k=1}^K{V}_{3k}+n{h}_n^{p/2}{\widehat{\delta }}^{-1}{K}^{-1/2}\sum _{k=1}^K{V}_{4k}\hfill \\ \hfill & ={\overline{V}}_1-2{\overline{V}}_2-2{\overline{V}}_3+{\overline{V}}_4\hfill \end{array}$$

Lemma 1.

Given Assumptions 1–5, under the null hypothesis, ${\delta }^2$ can be consistently estimated by ${\widehat{\delta }}^2\left(h_n\right)$, as $n{h}_n^p\to \infty$, $h_n\to 0$, $K\to \infty$.

Lemma 2.

Given Assumptions 1–5,

1. ${\overline{V}}_2=O_p\left(\sqrt{n}{h}_n^{p/2}\sqrt{E\left({\mathsf{\ell }}_N^2\left(\mathbf{x}\right)\right)}\right)$ uniformly for $m(·)$ under $H_1$, as $h_n\to 0$, $n{h}_n^p\to \infty$, $K\to \infty$.

2. ${\overline{V}}_3=O_p(h_n^{p/2}/\sqrt{K})=o_p\left(1\right)$, uniformly for $m(·)$ under $H_1$ and $H_0$, as $h_n\to 0$, $n{h}_n^p\to \infty$, $K\to \infty$.

3. ${\overline{V}}_4=O_p(h_n^{p/2}/\sqrt{K})=o_p\left(1\right)$, uniformly for $m(·)$ under $H_1$ and $H_0$, as $h_n\to 0$, $n{h}_n^p\to \infty$, $K\to \infty$.

Lemma 3.

Denote ${\tilde{V}}_1=n{h}_n^{p/2}{K}^{-1/2}{\sum }_{k=1}^K{V}_{1k}$. Under Assumptions 1–6, for any $m_N(·)\in H_1\left({\kappa }_N{\rho }_N\right)$ and n large enough, where ${\rho }_N=N^{-\frac{2s}{4s+p}}$, we have

$$E\left({\tilde{V}}_1\right)\le C_1{K}^{1/2}n{h}_n^{p/2}E{\mathsf{\ell }}_N^2\left(\mathbf{X}\right),Var\left({\tilde{V}}_1\right)\le C_{2}n{h}_n^{p}E{\mathsf{\ell }}_N^2\left(\mathbf{x}\right)+C_3$$

When

$${\kappa }_N{\rho }_N\ge \left(\left({\Lambda }_n+\sqrt{r\left(P_{h_n}\right)}\right)/\left({\Lambda }_n-{\displaystyle \frac{1}{\sqrt{n{h}_n^p}}}\right)\right)C_{0}L{h}_n^s,{\Lambda }_n={\displaystyle \frac{\sqrt{E\left({\pi }_k^{\prime }{P}_{h_n}{\pi }_k\right)}}{\sqrt{E\left({\pi }_k^{\prime }{\pi }_k\right)}}}$$

we have

$$E\left({\tilde{V}}_1\right)\ge C_1{K}^{1/2}n{h}_n^{p/2}{\left(\left({\Lambda }_n-{\displaystyle \frac{1}{\sqrt{n{h}_n^p}}}\right)\sqrt{E{\mathsf{\ell }}_N^2\left(\mathbf{x}\right)}-({\Lambda }_n+\sqrt{r\left(P_{h_n}\right)})C_{0}L{h}_n^s\right)}^2$$

Proof of Theorem 2.

We first construct a alternative $m_N(·)$ based on ${\tilde{h}}_n$.

$$m_N(·)=g(·;{\mathit{\theta }}^{*})+{\mathsf{\ell }}_N(·)$$

Using the method in Guerre and Lavergne [13] to construct the alternatives ${\mathsf{\ell }}_N(·)$, define

$$I_{\mathbf{t}}=\prod _{i=1}^p[t_i{\tilde{h}}_n,(t_i+1){\tilde{h}}_n)$$

for $\mathbf{t}\in {\mathrm{Y}}_n$.

$${\mathrm{Y}}_n=\{\mathbf{t}:\mathbf{t}={(t_1,\cdots ,t_p)}^{\prime }\in {\mathbb{N}}^p,0\le t_i\le {\gamma }_n-1\}$$

Then, $I_{\mathbf{t}}\subset {[0,1]}^p$, without loss of generality, we assume that ${\gamma }_n=1/{\tilde{h}}_n$ is an integer. Let

$${\phi }_{\mathbf{t}}\left(\mathbf{x}\right)=\phi \left({\displaystyle \frac{\mathbf{x}-\mathbf{t}{\tilde{h}}_n}{{\tilde{h}}_n}}\right),\mathbf{t}\in {\mathrm{Y}}_n$$

where ${\phi }_{\mathbf{t}}{(·)}^{\prime }s$ are orthogonal with disjoint supports $I_{\mathbf{t}}$, and $\phi (·)$ is bounded and nonnegative. Let $\{B_{\mathbf{t}},\mathbf{t}\in {\mathrm{Y}}_n\}$ be any sequence with $|B_{\mathbf{t}}|=1,\forall \mathbf{t}$,

$${\mathsf{\ell }}_N\left(\mathbf{x}\right)={\kappa }_N{\rho }_N\sum _{\mathbf{t}\in {\mathrm{Y}}_n}{B}_{\mathbf{t}}{\phi }_{\mathbf{t}}\left(\mathbf{x}\right)={\kappa }_N{\rho }_N\sum _{\mathbf{t}\in {\mathrm{Y}}_n}{B}_{\mathbf{t}}\phi \left({\displaystyle \frac{\mathbf{x}-\mathbf{t}{\tilde{h}}_n}{{\tilde{h}}_n}}\right)$$

Under Assumption 3, we have that there exists a constant C such that $E\left\{m_N^4\left(\mathbf{X}\right)\right\}\le C<\infty$ and $m_N(·)\in H_1\left({\kappa }_N{\rho }_N\right)$. Since

$$\underset{m(·)\in H_1\left({\kappa }_N{\rho }_N\right)}{inf}P\left\{n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)\ge z_{\alpha }\right\}\le P\left\{n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)\ge z_{\alpha }\right\}$$

for any $m(·)\in H_1\left({\kappa }_N{\rho }_N\right)$. The main idea is that if $I\{n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)\ge z_{\alpha }\}$ cannot be consistent against alternatives $m_N(·)$ as ${\kappa }_N\to \infty$, then we can conclude that it also not consistent uniformly against $m_N(·)\in H_1\left({\kappa }_N{\rho }_N\right)$ as ${\kappa }_N\to \infty$.

Based on Lemma 1–3, the test statistic

$$\begin{array}{cc}\hfill n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)& =n{h}_n^{p/2}\widehat{\delta }{\left(h_n\right)}^{-1}{K}^{-1/2}\sum _{k=1}^K{V}_k\hfill \\ \hfill & =\widehat{\delta }{\left(h_n\right)}^{-1}{\tilde{V}}_1-2n{h}_n^{p/2}\widehat{\delta }{\left(h_n\right)}^{-1}{K}^{-1/2}\sum _{k=1}{V}_{2k}+o_p\left(1\right)\hfill \\ \hfill & =O_p\left(1\right)E\left({\tilde{V}}_1\right)+O_p\left(\sqrt{Var\left({\tilde{V}}_1\right)}\right)-2{\overline{V}}_2+o_p\left(1\right)\hfill \end{array}$$

Without loss of generality, we assume that $\mathcal{K}(·)$ has support ${[-1,1]}^p$ in the following proofs:

(i): For $h_n=a_N{\tilde{h}}_n,a_N\to \infty$.

$$\begin{array}{cc}\hfill E\left({\tilde{V}}_1\right)& =\sqrt{K}n{h}_n^{p/2}E\left(V_{1k}\right)\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}E\left({\displaystyle \frac{1}{h_n^p}}\mathcal{K}\left({\displaystyle \frac{X_1-X_2}{h_n}}\right)\mathsf{\ell }\left(X_1\right)\mathsf{\ell }\left(X_2\right)\right)\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\int }_0^1{\int }_{(0,1)\cap ({\mathbf{x}}_2-\mathbf{1}{h}_n,{\mathbf{x}}_2+\mathbf{1}{h}_n)}{\displaystyle \frac{1}{h_n^p}}\mathcal{K}\left({\displaystyle \frac{{\mathbf{x}}_1-{\mathbf{x}}_2}{h_n}}\right)\mathsf{\ell }\left({\mathbf{x}}_1\right)\mathsf{\ell }\left({\mathbf{x}}_2\right)f\left({\mathbf{x}}_1\right)f\left({\mathbf{x}}_2\right)d{\mathbf{x}}_{1}d{\mathbf{x}}_2\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2{\displaystyle \frac{1}{h_n^p}}\sum _{\mathbf{t}\in {\mathrm{Y}}_n}{\int }_{\mathbf{t}{\tilde{h}}_n}^{(\mathbf{t}+\mathbf{1}){\tilde{h}}_n}{\int }_{(\mathbf{t}{\tilde{h}}_n,(\mathbf{t}+\mathbf{1}){\tilde{h}}_n)\cap ({\mathbf{x}}_2-\mathbf{1}{h}_n,{\mathbf{x}}_2+\mathbf{1}{h}_n)}\hfill \\ \hfill & \mathcal{K}\left({\displaystyle \frac{{\mathbf{x}}_1-{\mathbf{x}}_2}{h_n}}\right)\phi \left({\displaystyle \frac{{\mathbf{x}}_1-\mathbf{t}{\tilde{h}}_n}{{\tilde{h}}_n}}\right)\phi \left({\displaystyle \frac{{\mathbf{x}}_2-\mathbf{t}{\tilde{h}}_n}{{\tilde{h}}_n}}\right)f\left({\mathbf{x}}_1\right)f\left({\mathbf{x}}_2\right)d{\mathbf{x}}_{1}d{\mathbf{x}}_2\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2{\displaystyle \frac{1}{h_n^p}}\sum _{\mathbf{t}\in {\mathrm{Y}}_n}{\int }_{\mathbf{t}{\tilde{h}}_n}^{(\mathbf{t}+\mathbf{1}){\tilde{h}}_n}{\int }_{\mathbf{t}{\tilde{h}}_n}^{(\mathbf{t}+\mathbf{1}){\tilde{h}}_n}\mathcal{K}\left({\displaystyle \frac{{\mathbf{x}}_1-{\mathbf{x}}_2}{h_n}}\right)\hfill \\ \hfill & \phi \left({\displaystyle \frac{{\mathbf{x}}_1-\mathbf{t}{\tilde{h}}_n}{{\tilde{h}}_n}}\right)\phi \left({\displaystyle \frac{{\mathbf{x}}_2-\mathbf{t}{\tilde{h}}_n}{{\tilde{h}}_n}}\right)f\left({\mathbf{x}}_1\right)f\left({\mathbf{x}}_2\right)d{\mathbf{x}}_{1}d{\mathbf{x}}_2\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2{\displaystyle \frac{{\tilde{h}}_n^{2p}}{h_n^p}}\sum _{\mathbf{t}\in {\mathrm{Y}}_n}{\int }_0^1{\int }_0^1\mathcal{K}\left({\displaystyle \frac{(\mathbf{u}-\mathbf{v}){\tilde{h}}_n}{h_n}}\right)\phi \left(\mathbf{u}\right)\phi \left(\mathbf{v}\right)\hfill \\ \hfill & f(\mathbf{u}{\tilde{h}}_n+\mathbf{t}{\tilde{h}}_n)f(\mathbf{v}{\tilde{h}}_n+\mathbf{t}{\tilde{h}}_n)d\mathbf{u}d\mathbf{v}\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2{\displaystyle \frac{{\tilde{h}}_n^{2p}}{h_n^p}}\sum _{\mathbf{t}\in {\mathrm{Y}}_n}{\int }_0^1{\int }_0^1\mathcal{K}\left({\displaystyle \frac{\mathbf{u}-\mathbf{v}}{a_N}}\right)\phi \left(\mathbf{u}\right)\phi \left(\mathbf{v}\right)\hfill \\ \hfill & f(\mathbf{u}{\tilde{h}}_n+\mathbf{t}{\tilde{h}}_n)f(\mathbf{v}{\tilde{h}}_n+\mathbf{t}{\tilde{h}}_n)d\mathbf{u}d\mathbf{v}\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2{\displaystyle \frac{{\tilde{h}}_n^p}{h_n^p}}{\left({\tilde{h}}_n{\gamma }_n\right)}^p{\left[{\int }_0^1\phi \left(\mathbf{u}\right)f(\mathbf{u}{\tilde{h}}_n+\mathbf{t}{\tilde{h}}_n)d\mathbf{u}\right]}^2\hfill \\ \hfill & =O\left({\displaystyle \frac{{\kappa }_N^2}{K^{1/2}{a}_N^{p/2}}}\right)\hfill \end{array}$$

(ii): For $h_n={\tilde{h}}_n/a_N,a_N\to \infty$.

$$\begin{array}{cc}\hfill E\left({\tilde{V}}_1\right)& =\sqrt{K}n{h}_n^{p/2}E\left(V_{1k}\right)\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\int }_0^1{\int }_{(0,1)\cap ({\mathbf{x}}_2-\mathbf{1}{h}_n,{\mathbf{x}}_2+\mathbf{1}{h}_n)}{\displaystyle \frac{1}{h_n^p}}\mathcal{K}\left({\displaystyle \frac{{\mathbf{x}}_1-{\mathbf{x}}_2}{h_n}}\right)\mathsf{\ell }\left({\mathbf{x}}_1\right)\mathsf{\ell }\left({\mathbf{x}}_2\right)f\left({\mathbf{x}}_1\right)f\left({\mathbf{x}}_2\right)d{\mathbf{x}}_{1}d{\mathbf{x}}_2\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2{\displaystyle \frac{1}{h_n^p}}\sum _{\mathbf{t}\in {\mathrm{Y}}_n}{\int }_{\mathbf{t}{\tilde{h}}_n}^{(\mathbf{t}+\mathbf{1}){\tilde{h}}_n}{\int }_{(\mathbf{t}{\tilde{h}}_n,(\mathbf{t}+\mathbf{1}){\tilde{h}}_n)\cap ({\mathbf{x}}_2-\mathbf{1}{h}_n,{\mathbf{x}}_2+\mathbf{1}{h}_n)}\hfill \\ \hfill & \mathcal{K}\left({\displaystyle \frac{{\mathbf{x}}_1-{\mathbf{x}}_2}{h_n}}\right)\phi \left({\displaystyle \frac{{\mathbf{x}}_1-\mathbf{t}{\tilde{h}}_n}{{\tilde{h}}_n}}\right)\phi \left({\displaystyle \frac{{\mathbf{x}}_2-\mathbf{t}{\tilde{h}}_n}{{\tilde{h}}_n}}\right)f\left({\mathbf{x}}_1\right)f\left({\mathbf{x}}_2\right)d{\mathbf{x}}_{1}d{\mathbf{x}}_2\hfill \\ \hfill & \le \sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2{\tilde{h}}_n^p\sum _{\mathbf{t}\in {\mathrm{Y}}_n}{\int }_0^1{\int }_{-1}^1\mathcal{K}\left(u\right)\phi \left(\mathbf{u}·{\displaystyle \frac{h_n}{{\tilde{h}}_n}}+\mathbf{v}\right)\phi \left(\mathbf{v}\right)\hfill \\ \hfill & f(\mathbf{u}{h}_n+\mathbf{t}{\tilde{h}}_n+\mathbf{v}{\tilde{h}}_n)f(\mathbf{v}{\tilde{h}}_n+\mathbf{t}{\tilde{h}}_n)d\mathbf{u}d\mathbf{v}\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2{\left({\tilde{h}}_n{\gamma }_n\right)}^p{\int }_0^1{\int }_{-1}^1\mathcal{K}\left(u\right)\phi \left(\mathbf{u}·{\displaystyle \frac{h_n}{{\tilde{h}}_n}}+\mathbf{v}\right)\phi \left(\mathbf{v}\right)\hfill \\ \hfill & f(\mathbf{u}{h}_n+\mathbf{t}{\tilde{h}}_n+\mathbf{v}{\tilde{h}}_n)f(\mathbf{v}{\tilde{h}}_n+\mathbf{t}{\tilde{h}}_n)d\mathbf{u}d\mathbf{v}\hfill \\ \hfill & =\sqrt{K}n{h}_n^{p/2}{\kappa }_N^2{\rho }_N^2\hfill \\ \hfill & =O\left({\displaystyle \frac{{\kappa }_N^2}{K^{1/2}{a}_N^{p/2}}}\right)\hfill \end{array}$$

Through tedious calculation, we can also get $\sqrt{Var\left({\tilde{V}}_1\right)}=O\left(1\right)$ and ${\overline{V}}_2=o_p\left(1\right)$ as $h_n\to 0$ for above two cases. Therefore, for any $a_N$, there exists ${\kappa }_N$ such that $n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)=O_p\left(1\right)$ as $h_n\to 0$. Therefore, we cannot get $P\left\{n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)\ge z_{\alpha }\right\}\to 1$. We obtain the same conclusion for $\underset{m(·)\in H_1\left({\kappa }_N{\rho }_N\right)}{inf}P\left\{n{h}_n^{p/2}{K}^{1/2}{T}_N\left(h_n\right)\ge z_{\alpha }\right\}$. Thus, the theorem is proved. □