On the variable bandwidth kernel estimation of conditional U-statistics at optimal rates in sup-norm

• Published in: Physica A Vol. 625; p. 129000
• Main Authors: Bouzebda, Salim, Taachouche, Nourelhouda
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2023; Elsevier
• Subjects: Bias reduction; Empirical process methods; Functional Analysis; Kernel estimation; Machine learning problems; Mathematics; Nonparametric estimation; Probability; Regression-type models; Statistics; Statistics Theory; U-process; U-statistics; Uniform in bandwidth; Variable bandwidth; 62G05; 62G07; 60G10; Nonparametric estimation; Empirical process methods; Kernel estimation; U-process; Machine learning problems; 62H 12; Regression-type models; U-statistics; Bias reduction; Uniform in bandwidth; 60F05; 60G15; Variable bandwidth
• ISSN: 0378-4371; 0378-4371
• DOI: 10.1016/j.physa.2023.129000

U-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. U-statistics generalize the empirical mean of a random variable X to sums over every k-tuple of distinct observations of X. W. Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya–Watson estimates of a regression function. Stute proved their strong pointwise consistency to: r(k)(φ,t)=E[φ(Y1,…,Yk)|(X1,…,Xk)=t],fort∈Rpk.In this paper we introduce a variable bandwidth kernel conditional U-statistics estimator for i.i.d. observations in Rpq to improve the classical Stute estimator. We apply in nontrivial way the methods developed in Dony and Mason (2008) to establish uniform in t and in bandwidth consistency to r(k)(φ,t) of the proposed estimator. Our estimator includes density estimators proposed by McKay (1993) as a particular case. In addition, uniform consistency is also established over φ∈ℱ for a suitably restricted class ℱ, in both cases bounded and unbounded, satisfying some moment conditions. Our theorems allow data-driven local bandwidths for the considered estimators. Moreover, in the same context, we show the uniform bandwidth 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. The theoretical uniform consistency results established in this paper are (or will be) key tools for many further developments in regression analysis. These results are proved under some standard structural conditions on the Vapnik–Chervonenkis classes of functions and some mild conditions on the model. The applications of our main results include, among many others, the discrimination problems, the metric learning, the Kendall rank correlation coefficient, the generalized U-Statistics and the set indexed conditional U-statistics. Finally, the finite-sample performance of the proposed method is investigated in a simulation study.