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, France; mokkadem@math.uvsq.fr.pelletier@math.uvsq.fr.

Received: 1 March 2002
Revised: 2 May 2002

Abstract

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