On local $U$-statistic processes and the estimation of densities of functions of several sample variables

Details On local $U$-statistic processes and the estimation of densities of functions of several sample variables On local $U$-statistic processes and the estimation of densities of functions of several sample variables

2007-08-16

arxiv.org/abs/0708.2227v1

Annals of Statistics 2007, Vol. 35, No. 3, 1105-1145

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053607000000154 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053607000000154

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 $L_p$ norms are obtained for these estimators.

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