This paper introduces a new stochastic process, a collection of $U$-statistics indexed by a family of symmetric kernels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation in density estimation is given. The proofs adapt methods from the theory of empirical processes.
Publié le : 1987-06-14
Classification:
$U$-statistics,
empirical processes,
rates of convergence,
cross-validation,
reversed submartingale,
maximal inequality,
kernel density estimation,
60F15,
62G99,
60G20
@article{1176350374,
author = {Nolan, Deborah and Pollard, David},
title = {$U$-Processes: Rates of Convergence},
journal = {Ann. Statist.},
volume = {15},
number = {1},
year = {1987},
pages = { 780-799},
language = {en},
url = {http://dml.mathdoc.fr/item/1176350374}
}
Nolan, Deborah; Pollard, David. $U$-Processes: Rates of Convergence. Ann. Statist., Tome 15 (1987) no. 1, pp. 780-799. http://gdmltest.u-ga.fr/item/1176350374/