Search Results (10)

U-statistics are fundamental in modeling statistical measures that involve responses from multiple subjects. They generalize the concept of the empirical mean of a random variable X to include summations over each m-tuple of distinct observations of X. W. Stute introduced conditional U-statistics, extending the Nadaraya–Watson estimates for regression functions. Stute demonstrated their strong pointwise consistency with the conditional expectation $r^{\left(m\right)}(\phi ,\mathbf{t})$, defined as $\mathbb{E}\left[\phi (Y_1,\dots ,Y_m)\right|(X_1,\dots ,X_m)=\mathbf{t}]$ for $\mathbf{t}\in {\mathcal{X}}^m$. This paper focuses on estimating functional single index (FSI) conditional U-processes for regular time series data. We propose a novel, automatic, and location-adaptive procedure for estimating these processes based on k-Nearest Neighbor (kNN) principles. Our asymptotic analysis includes data-driven neighbor selection, making the method highly practical. The local nature of the kNN approach improves predictive power compared to traditional kernel estimates. Additionally, we establish new uniform results in bandwidth selection for kernel estimates in FSI conditional U-processes, including almost complete convergence rates and weak convergence under general conditions. These results apply to both bounded and unbounded function classes, satisfying certain moment conditions, and are proven under standard Vapnik–Chervonenkis structural conditions and mild model assumptions. Furthermore, we demonstrate uniform consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship. This result is independently valuable and has potential applications in areas such as set-indexed conditional U-statistics, the Kendall rank correlation coefficient, and discrimination problems. Full article

As a novel learning algorithm for feedforward neural networks, the twin extreme learning machine (TELM) boasts advantages such as simple structure, few parameters, low complexity, and excellent generalization performance. However, it employs the squared $L_2$-norm metric and an unbounded hinge loss function, which tends to overstate the influence of outliers and subsequently diminishes the robustness of the model. To address this issue, scholars have proposed the bounded capped $L_{2,p}$-norm metric, which can be flexibly adjusted by varying the p value to adapt to different data and reduce the impact of noise. Therefore, we substitute the metric in the TELM with the capped $L_{2,p}$-norm metric in this paper. Furthermore, we propose a bounded, smooth, symmetric, and noise-insensitive squared fractional loss (SF-loss) function to replace the hinge loss function in the TELM. Additionally, the TELM neglects statistical information in the data; thus, we incorporate the Fisher regularization term into our model to fully exploit the statistical characteristics of the data. Drawing upon these merits, a squared fractional loss-based robust supervised twin extreme learning machine (SF-RSTELM) model is proposed by integrating the capped $L_{2,p}$-norm metric, SF-loss, and Fisher regularization term. The model shows significant effectiveness in decreasing the impacts of noise and outliers. However, the proposed model’s non-convexity poses a formidable challenge in the realm of optimization. We use an efficient iterative algorithm to solve it based on the concave-convex procedure (CCCP) algorithm and demonstrate the convergence of the proposed algorithm. Finally, to verify the algorithm’s effectiveness, we conduct experiments on artificial datasets, UCI datasets, image datasets, and NDC large datasets. The experimental results show that our model is able to achieve higher ACC and $F_1$ scores across most datasets, with improvements ranging from 0.28% to 4.5% compared to other state-of-the-art algorithms. Full article

In capture–recapture experiments, the presence of overdispersion and heterogeneity necessitates the use of the negative binomial regression model for inferring population sizes. However, within this model, existing methods based on likelihood and ratio regression for estimating the dispersion parameter often face boundary and nonidentifiability issues. These problems can result in nonsensically large point estimates and unbounded upper limits of confidence intervals for the population size. We present a penalized empirical likelihood technique for solving these two problems by imposing a half-normal prior on the population size. Based on the proposed approach, a maximum penalized empirical likelihood estimator with asymptotic normality and a penalized empirical likelihood ratio statistic with asymptotic chi-square distribution are derived. To improve numerical performance, we present an effective expectation-maximization (EM) algorithm. In the M-step, optimization for the model parameters could be achieved by fitting a standard negative binomial regression model via the R basic function glm.nb(). This approach ensures the convergence and reliability of the numerical algorithm. Using simulations, we analyze several synthetic datasets to illustrate three advantages of our methods in finite-sample cases: complete mitigation of the boundary problem, more efficient maximum penalized empirical likelihood estimates, and more precise penalized empirical likelihood ratio interval estimates compared to the estimates obtained without penalty. These advantages are further demonstrated in a case study estimating the abundance of black bears (Ursus americanus) at the U.S. Army’s Fort Drum Military Installation in northern New York. Full article

In his work published in (Ann. Probab. 19, No. 2 (1991), 812–825), W. Stute introduced the notion of conditional U-statistics, expanding upon the Nadaraya–Watson estimates used for regression functions. Stute illustrated the pointwise consistency and asymptotic normality of these statistics. Our research extends these concepts to a broader scope, establishing, for the first time, an asymptotic framework for single-index conditional U-statistics applicable to locally stationary random fields $\{{\mathbf{X}}_{s,A_n}:sin{R}_n\}$ observed at irregularly spaced locations in $R_n$, a subset of ${\mathbb{R}}^d$. We introduce an estimator for the single-index conditional U-statistics operator that accommodates the nonstationary nature of the data-generating process. Our method employs a stochastic sampling approach that allows for the flexible creation of irregularly spaced sampling sites, covering both pure and mixed increasing domain frameworks. We establish the uniform convergence rate and weak convergence of the single conditional U-processes. Specifically, we examine weak convergence under bounded or unbounded function classes that satisfy specific moment conditions. These findings are established under general structural conditions on the function classes and underlying models. The theoretical advancements outlined in this paper form essential foundations for potential breakthroughs in functional data analysis, laying the groundwork for future research in this field. Moreover, in the same context, we show the uniform consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship, which is of its own interest. Potential applications of our findings encompass, among many others, the set-indexed conditional U-statistics, the Kendall rank correlation coefficient, and the discrimination problems. Full article

In financial and statistical computations, calculating expectations often requires evaluating integrals with respect to a Gaussian measure. Monte Carlo methods are widely used for this purpose due to their dimension-independent convergence rate. Quasi-Monte Carlo is the deterministic analogue of Monte Carlo and has the potential to substantially enhance the convergence rate. Importance sampling is a widely used variance reduction technique. However, research into the specific impact of importance sampling on the integrand, as well as the conditions for convergence, is relatively scarce. In this study, we combine the randomly shifted lattice rule with importance sampling. We prove that, for unbounded functions, randomly shifted lattice rules combined with a suitably chosen importance density can achieve convergence as quickly as $O\left(N^{-1+ϵ}\right)$, given N samples for arbitrary $ϵ$ values under certain conditions. We also prove that the conditions of convergence for Laplace importance sampling are stricter than those for optimal drift importance sampling. Furthermore, using a generalized linear mixed model and Randleman–Bartter model, we provide the conditions under which functions utilizing Laplace importance sampling achieve convergence rates of nearly $O\left(N^{-1+ϵ}\right)$ for arbitrary $ϵ$ values. Full article

In the literature, the use of fractional moments to express the available information in the framework of maximum entropy (MaxEnt) approximation of a distribution F having finite or unbounded positive support, has been essentially considered as a computational tool to improve the performance of the analogous procedure based on integer moments. No attention has been paid to two formal aspects concerning fractional moments, such as conditions for the existence of the maximum entropy approximation based on them or convergence in entropy of this approximation to F. This paper aims to fill this gap by providing proofs of these two fundamental results. In fact, convergence in entropy can be involved in the optimal selection of the order of fractional moments for accelerating the convergence of the MaxEnt approximation to F, to clarify the entailment relationships of this type of convergence with other types of convergence useful in statistical applications, and to preserve some important prior features of the underlying F distribution. Full article

Stute presented the so-called conditional U-statistics generalizing the Nadaraya–Watson estimates of the regression function. Stute demonstrated their pointwise consistency and the asymptotic normality. In this paper, we extend the results to a more abstract setting. We develop an asymptotic theory of conditional U-statistics for locally stationary random fields $\{{\mathit{X}}_{s,A_n}:s\mathrm{in}{R}_n\}$ observed at irregularly spaced locations in $R_n={[0,A_n]}^d$ as a subset of ${\mathbb{R}}^d$. We employ a stochastic sampling scheme that may create irregularly spaced sampling sites in a flexible manner and includes both pure and mixed increasing domain frameworks. We specifically examine the rate of the strong uniform convergence and the weak convergence of conditional U-processes when the explicative variable is functional. We examine the weak convergence where the class of functions is either bounded or unbounded and satisfies specific moment conditions. These results are achieved under somewhat general structural conditions pertaining to the classes of functions and the underlying models. The theoretical results developed in this paper are (or will be) essential building blocks for several future breakthroughs in functional data analysis. Full article

Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. For applications, we introduce so-called statistical-unbounded convergence and use these convergences to describe KB-spaces and reflexive Banach lattices. Full article

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented. Full article

Branching network is one of the most universal phenomena in living or non-living systems, such as river systems and the bronchial trees of mammals. To topologically characterize the branching networks, the Branch Length Similarity (BLS) entropy was suggested and the statistical methods based on the entropy have been applied to the shape identification and pattern recognition. However, the mathematical properties of the BLS entropy have not still been explored in depth because of the lack of application and utilization requiring advanced mathematical understanding. Regarding the mathematical study, it was reported, as a theorem, that all BLS entropy values obtained for simple networks created by connecting pixels along the boundary of a shape are exactly unity when the shape has infinite resolution. In the present study, we extended the theorem to the network created by linking infinitely many nodes distributed on the bounded or unbounded domain in Rk for k ≥ 1. We proved that all BLS entropies of the nodes in the network go to one as the number of nodes, n, goes to infinite and its convergence rate is 1 - O(1= ln n), which was confirmed by the numerical tests. Full article