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

Mokkadem, Abdelkader; Pelletier, Mariane

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

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 $l^p$ 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.

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