On the weak convergence and the uniform-in-bandwidth consistency of the general conditional $U$-processes based on the copula representation: multivariate setting

Abstract

$U$-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. $U$-statistics generalise the empirical mean of a random variable $X$ to sums over every $m$-tuple of distinct observations of $X$. Stute [Conditional U -statistics, Ann. Probab., 1991] introduced a class of estimators called conditional $U$-statistics. In the present work, we provide a new class of estimators of conditional $U$-statistics. More precisely, we investigate the conditional $U$-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. In addition, uniform consistency is also established over $\varphi \in \mathscr{F}$ for a suitably restricted class $\mathscr{F}$, 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 estimators of the regression function under random censorship, which is of its own interest. We also consider the weak convergence of the conditional $U$-statistics processes. We discuss the wild bootstrap of the conditional $U$-statistics processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis class of functions and some mild conditions on the model.