Estimation of a regular conditional functional by conditional U‐statistic regression

Abstract

U‐statistics constitute a large class of estimators, generalizing the empirical mean of a random variable  to sums over every ‐tuple of distinct observations of . They may be used to estimate a regular functional of the law of . When a vector of covariates is available, a conditional U‐statistic describes the effect of on the conditional law of given , by estimating a regular conditional functional . We state nonasymptotic bounds of general conditional U‐statistics and study their asymptotics too. Assuming a parametric model of the conditional functional of interest, we propose a regression‐type estimator based on conditional U‐statistics. Its theoretical properties are derived, first in a nonasymptotic framework and then in two different asymptotic regimes. Some examples are given to illustrate our methods.