Let $f$ be a density on the real line and let $f_n$ be the kernel estimate of $f$ in which the smoothing factor is obtained by maximizing the cross-validated likelihood product according to the method of Duin and Habbema, Hermans and Vandenbroek. Under mild regularity conditions on the kernel and $f$, we show, among other things that $\int|f_n - f| \rightarrow 0$ almost surely if and only if the sample extremes of $f$ are strongly stable.
Publié le : 1989-09-14
Classification:
Density function,
nonparametric estimation,
consistency,
order statistics,
strong stability,
kernel estimate,
62G05
@article{1176347256,
author = {Broniatowski, Michel and Deheuvels, Paul and Devroye, Luc},
title = {On the Relationship Between Stability of Extreme Order Statistics and Convergence of the Maximum Likelihood Kernel Density Estimate},
journal = {Ann. Statist.},
volume = {17},
number = {1},
year = {1989},
pages = { 1070-1086},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347256}
}
Broniatowski, Michel; Deheuvels, Paul; Devroye, Luc. On the Relationship Between Stability of Extreme Order Statistics and Convergence of the Maximum Likelihood Kernel Density Estimate. Ann. Statist., Tome 17 (1989) no. 1, pp. 1070-1086. http://gdmltest.u-ga.fr/item/1176347256/