Asymptotic properties of conditional U-statistics using delta sequences

Abstract

Stute (1991) introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: r ( k ) ( φ , t ) : = E [ φ ( Y 1 , … , Y k ) | ( X 1 , … , X k ) = t ] , for t ∈ R d k . This article deals with a quite general non parametric statistical curve estimation setting including the Stute estimator as a particular case. The class of “delta sequence estimators” is defined and treated here. This class includes also the orthogonal series and histogram methods. The theoretical results concerning the exponential inequalities and the asymptotic normality, established in this article, are (or will be) key tools for many further developments in functional estimation. As a by-product of our proofs, we state consistency results for the delta sequences conditional U-statistics estimator, under the random censoring. Potential applications include discrimination problems, metric learning and multipartite ranking, Kendall rank correlation coefficient, generalized U-statistics, and set indexed conditional U-statistics.