Uniform in bandwidth consistency of conditional U-statistics

Abstract

Stute [Ann. Probab. 19 (1991) 812-825] introduced a class of estimators called conditional U-statistics. They can be seen as a generalization of the Nadaraya-Watson estimator for the regression function. Stute proved their strong pointwise consistency to m(t) := E[g(Yl, ⋯, Ym) (Xl, ⋯, Xm)=t], t∈ℝm. Very recently, Giné and Mason introduced the notion of a local U-process, which generalizes that of a local empirical process, and obtained central limit theorems and laws of the iterated logarithm for this class. We apply the methods developed in Einmahl and Mason [Ann. Statist. 33 (2005) 1380-1403] and Giné and Mason [Ann. Statist. 35 (2007) 1105-1145; J. Theor Probab. 20 (2007) 457-485] to establish uniform in t and in bandwidth consistency to m(t) of the estimator proposed by Stute. We also discuss how our results are used in the analysis of estimators with data-dependent bandwidths. © 2008 ISI/BS.