Estimation of a regular conditional functional by conditional U-statistics regression
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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(P_X) of the law of X. When a vector of covariates Z is available, a conditional U-statistic may describe the effect of z on the conditional law of X given Z=z, by estimating a regular conditional functional θ(P_X|Z=·). We prove concentration inequalities for conditional U-statistics. 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 non-asymptotic framework and then in two different asymptotic regimes. Some examples are given to illustrate our methods.
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