The law of the iterated logarithm for the multivariate kernel mode estimator
ESAIM: Probability and Statistics, Volume 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, $p\in \left[1,\infty \right]$, 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.

DOI: 10.1051/ps:2003004
Classification: 62G05, 62G20, 60F05, 60F15
Keywords: 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, Volume 7 (2003), pp. 1-21. doi : 10.1051/ps:2003004. http://www.numdam.org/articles/10.1051/ps:2003004/

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