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The law of the iterated logarithm for the multivariate kernel modeestimator

Published online by Cambridge University Press:  15 May 2003

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

Let θ be the mode of a probability density and θn 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 θn - 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 θn - θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the lp norms, p ∈ [1,∞], of θn - θ. Finally, we consider the case θ is degenerate and give the exact weak and strong convergence rate of θn - θ in the univariate framework.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2003

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