Uniform in bandwidth consistency of conditional U-statistics

In 1991 Stute 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, and Stute proved their strong pointwise consistency to the generalized regression function m(t):=E[g(Y_1,...,Y_m)|(X_1,...,X_m)=t], with t in 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 (2005) and Giné and Mason (2007a,b) to establish uniform in bandwidth consistency to m(t) of the estimator proposed by Stute. We discuss how our results are used in the analysis of estimators with data dependent bandwidth.