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

• Volume: 7, page 1-21
• ISSN: 1292-8100

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## Abstract

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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.

## How to cite

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Mokkadem, Abdelkader, and Pelletier, Mariane. "The law of the iterated logarithm for the multivariate kernel mode estimator." ESAIM: Probability and Statistics 7 (2003): 1-21. <http://eudml.org/doc/244798>.

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 $l^p$ 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.},
journal = {ESAIM: Probability and Statistics},
keywords = {density; mode; kernel estimator; central limit theorem; law of the iterated logarithm},
language = {eng},
pages = {1-21},
publisher = {EDP-Sciences},
title = {The law of the iterated logarithm for the multivariate kernel mode estimator},
url = {http://eudml.org/doc/244798},
volume = {7},
year = {2003},
}

TY - JOUR
AU - Pelletier, Mariane
TI - The law of the iterated logarithm for the multivariate kernel mode estimator
JO - ESAIM: Probability and Statistics
PY - 2003
PB - EDP-Sciences
VL - 7
SP - 1
EP - 21
AB - 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.
LA - eng
KW - density; mode; kernel estimator; central limit theorem; law of the iterated logarithm
UR - http://eudml.org/doc/244798
ER -

## References

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