Uniform in Bandwidth Consistency of Conditional U-statistics Adaptive to Intrinsic Dimension in Presence of Censored Data

• Published in: Sankhya A Vol. 85; no. 2; pp. 1548 - 1606
• Main Authors: Bouzebda, Salim, El-hadjali, Thouria, Ferfache, Anouar Abdeldjaoued
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.08.2023; Springer Verlag
• Subjects: Functional Analysis; Mathematics; Mathematics and Statistics; Probability; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Statistics Theory; processes; Primary 60F05; 60G10; 62G08; conditional; functional estimation; conditional empirical processes; 62G07; Secondary 62G15; kernel estimation; VC-classes; regression; Non-parametric estimation; 62G30; 60G15
• ISSN: 0976-836X; 0972-7671; 0976-8378; 0976-836X
• DOI: 10.1007/s13171-022-00301-7

U -statistics represent a fundamental class of statistics from modelling quantities of interest defined by multi-subject responses. U -statistics generalize the empirical mean of a random variable X to sums over every m -tuple of distinct observations of Stute (Ann. Probab. 19 , 812–825 1991 ) 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 ( t ) : = E [ φ ( Y 1 , … , Y m ) | ( X 1 , … , X m ) = t ] , for t ∈ ℝ d m . We apply the methods developed in Dony and Mason (Bernoulli 14 (4), 1108–1133 2008 ) to establish uniform in t and in bandwidth consistency (i.e., h , h ∈ [ a n , b n ] where 0 < a n < b n → 0 at some specific rate) to r ( t ) of the estimator proposed by Stute when Y , under weaker conditions on the kernel than previously used in the literature. We extend existing uniform bounds on the kernel conditional U -statistic estimator and make it adaptive to the intrinsic dimension of the underlying distribution of X which the so-called intrinsic dimension will characterize. In addition, uniform consistency is also established over φ ∈ 𝒡 for a suitably restricted class 𝒡 , in both cases bounded and unbounded, satisfying some moment conditions. Our theorems allow data-driven local bandwidths for these statistics. 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.