Uniform consistency and uniform in bandwidth consistency for nonparametric regression estimates and conditional U-statistics involving functional data

Abstract

ABSTRACTW. Stute [(1991), Annals of Probability, 19, 812–825] introduced a class of so-called conditional U-statistics, which may be viewed as a generalisation of the Nadaraya–Watson estimates of a regression function. Stute proved their strong pointwise consistency to We apply the methods developed in Dony and Mason [(2008), Bernoulli, 14(4), 1108–1133] to establish uniformity in and in bandwidth consistency (i.e. , where at some specific rate) to of the estimator proposed by Stute when Y and covariates X are functional taking value in some abstract spaces. In addition, uniform consistency is also established over for a suitably restricted class . The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in functional data analysis. Applications include the Nadaraya–Watson kernel estimators and the conditional distribution function. Our theorems allow data-driven local bandwidths for these statistics.