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

U‐statistics constitute a large class of estimators, generalizing the empirical mean of a random variable X to sums over every k‐tuple of distinct observations of X. They may be used to estimate a regular functional θ(ℙX) of the law of X. When a vector of covariates Z is available, a conditional U‐statistic describes the effect of z on the conditional law of X given Z=z, by estimating a regular conditional functional θ(ℙX|Z=·). 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.