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 $$m(\mathbf{t}):=\mathbb{E}[g(Y_{1},\ldots,Y_{m})|(X_{1},\ldots,X_{m})=\mathbf{t}],\qquad\mathbf{t}\in\mathbb{R}^{m}.$$ 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.