Weak-convergence of empirical conditional processes and conditional U-processes involving functional mixing data

• Published in: Statistical inference for stochastic processes : an international journal devoted to time series analysis and the statistics of continuous time processes and dynamic systems Vol. 26; no. 1; pp. 33 - 88
• Main Authors: Bouzebda, Salim, Nemouchi, Boutheina
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2023; Springer Nature B.V
• Subjects: Convergence; Data analysis; Empirical analysis; Functionals; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Random variables; Statistical Theory and Methods; Statistics; 62G05; Secondary: 62E20; statistic; 62G07; 62G08; Kolmogorov’s entropy; Conditional; Kernel-type estimators; Empirical processes; VC-classes; Weak convergence; Functional regression; 62G30; 62G20; 62G32; Small ball probability; 62G35; Functional data analysis
• ISSN: 1387-0874; 1572-9311
• DOI: 10.1007/s11203-022-09276-6

U -statistics represent a fundamental class of statistics arising 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 m -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 : m ( t ) : = E [ φ ( Y 1 , … , Y m ) | ( X 1 , … , X m ) = t ] , for t ∈ X m . In this paper we are mainly interested in establishing weak convergence of conditional U -processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of conditional U -processes when the explicative variable is functional. We treat the weak convergence in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for many further developments in functional data analysis.