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

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 $L_p$ norms are obtained for these estimators.

arXiv commentPublished at http://dx.doi.org/10.1214/009053607000000154 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Exponential and moment inequalities for U-statistics
• Concentration inequalities and asymptotic results for ratio type empirical processes
• A uniform functional law of the logarithm for the local empirical process

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• Approximating high-dimensional infinite-order $U$-statistics: statistical and computational guarantees
• Uniform central limit theorems for the Grenander estimator
• Stratified incomplete local simplex tests for curvature of nonparametric multiple regression
• Uniform convergence of convolution estimators for the response density in nonparametric regression