Estimation of a regular conditional functional by conditional U-statistic regression
Alexis Derumigny (TU Delft - Statistics)
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Abstract
U-statistics constitute a large class of estimators, generalizing the empirical mean of a random variable (Formula presented.) to sums over every (Formula presented.) -tuple of distinct observations of (Formula presented.). They may be used to estimate a regular functional (Formula presented.) of the law of (Formula presented.). When a vector of covariates (Formula presented.) is available, a conditional U-statistic describes the effect of (Formula presented.) on the conditional law of (Formula presented.) given (Formula presented.), by estimating a regular conditional functional (Formula presented.). 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.
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