Volume 7, March 2003
|Page(s)||1 - 21|
|Published online||15 May 2003|
The law of the iterated logarithm for the multivariate kernel mode estimator
Département de Mathématiques,
Université de Versailles-Saint-Quentin,
45 avenue des États-Unis,
78035 Versailles Cedex,
Revised: 2 May 2002
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.
Mathematics Subject Classification: 62G05 / 62G20 / 60F05 / 60F15
Key words: Density / mode / kernel estimator / central limit theorem / law of the iterated logarithm.
© EDP Sciences, SMAI, 2003
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