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

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.