On local U-statistic processes and the estimation of densities of functions of several sample variables
The Annals of Statistics · 2007 · Volume 35 · Issue 3
DOI: 10.1214/009053607000000154
Abstract
A notion of local U-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for U-processes that may be useful in other contexts. This local U-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517–525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the Lp norms are obtained for these estimators.
Keywords
60F05, 60F15, 62E20, 62G30, central limit theorems, empirical process, kernel density estimation, U-statistics
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