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Issue ESAIM: P&S
Volume 7, 2003
Page(s) 1 - 21
DOI 10.1051/ps:2003004

ESAIM: P&S, March 2003, Vol. 7, pp. 1-21
DOI: 10.1051/ps:2003004

The law of the iterated logarithm for the multivariate kernel mode estimator

Abdelkader Mokkadem and Mariane Pelletier

Département de Mathématiques, Université de Versailles-Saint-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France; mokkadem@math.uvsq.fr.pelletier@math.uvsq.fr.


(Received March 1, 2002. Revised May 2, 2002)

Abstract
Let $\theta$ be the mode of a probability density and $\theta_n$ its kernel estimator. In the case $\theta$ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for $\theta_n-\theta$. Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence $\theta_n-\theta$ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the lp norms, $p\in[1,\infty]$, of $\theta_n-\theta$. Finally, we consider the case $\theta$ is degenerate and give the exact weak and strong convergence rate of $\theta_n-\theta$ in the univariate framework.


Mathematics Subject Classification. 62G05, 62G20, 60F05, 60F15

Key words: Density, mode, kernel estimator, central limit theorem, law of the iterated logarithm.


© EDP Sciences, SMAI 2003


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