Uniform in bandwidth consistency of conditional U-statistics

Stute [Anti. 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(Y(1), ..., Y(m))|(X(1), ..., X(m)) = t] t is an element of R(m). Very recently, Gine 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 Gine and Mason [Anti. Statist. 35 (2007) 1105-1145; J Theor Probab, 20 (2007) 457-485] to establish uniform in t and in bandwidth consistency to in(t) of the estimator proposed by Stute. We also discuss how Our results are used in the analysis of estimators with data-dependent bandwidths.