Uniform in bandwidth consistency of conditional statistics

摘要

Very recently, Gine and Mason introduced the notion of a local [/-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 [Ann Statist 35 (2007) 1105-1145;./. Theor. Probab. 20 (2007) 457^85] 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.