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

Mokkadem, Abdelkader; Pelletier, Mariane

doi:10.1051/ps:2003004

Mokkadem, Abdelkader ; Pelletier, Mariane

ESAIM: Probability and Statistics, Tome 7 (2003), pp. 1-21.

Résumé

Let $θ$ be the mode of a probability density and $θ_{n}$ its kernel estimator. In the case $θ$ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for $θ_{n} - θ$ . 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 $θ_{n} - θ$ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the $l^{p}$ norms, $p \in [1, \infty]$ , of $θ_{n} - θ$ . Finally, we consider the case $θ$ is degenerate and give the exact weak and strong convergence rate of $θ_{n} - θ$ in the univariate framework.

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/ps:2003004

Classification : 62G05, 62G20, 60F05, 60F15
Mots-clés : density, mode, kernel estimator, central limit theorem, law of the iterated logarithm

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Mokkadem, Abdelkader; Pelletier, Mariane. The law of the iterated logarithm for the multivariate kernel mode estimator. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 1-21. doi : 10.1051/ps:2003004. https://www.numdam.org/articles/10.1051/ps:2003004/

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