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The law of the iterated logarithmfor the multivariate kernel modeestimator

Published online by Cambridge University Press:  15 May 2003

Abdelkader Mokkadem  and
Mariane Pelletier
Abdelkader Mokkadem
Affiliation:
Département de Mathématiques, Université de Versailles-Saint-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France; [email protected]@math.uvsq.fr.
Mariane Pelletier
Affiliation:
Département de Mathématiques, Université de Versailles-Saint-Quentin, 45 avenue des États-Unis, 78035 Versailles Cedex, France; [email protected]@math.uvsq.fr.
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Abstract

Let θ be the mode of a probability density and θn itskernel estimator. In the case θ is nondegenerate, we first specify the weakconvergence rate of the multivariate kernel mode estimator by stating the central limittheorem for θn - θ. Then, we obtain a multivariate law ofthe iterated logarithm for the kernel mode estimator by proving that, with probabilityone, the limit set of the sequence θn - θ suitably normalized is an ellipsoid.We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], ofθn - θ. Finally, we consider the case θ is degenerate and give the exactweak and strong convergence rate of θn - θ in the univariate framework.

Keywords

Densitymodekernel estimatorcentral limit theoremlaw of the iterated logarithm.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 1 - 21
Copyright
© EDP Sciences, SMAI, 2003

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