Density estimation for associated sampling: a point process influenced approach

Abstract

Let Xn, n E¸ IN, be a sequence of associated variables with common density function. We study the kernel estimation of this density, based on the given sequence of variables. Sufficient conditions are given for the consistency and asymptotic normality of the kernel estimator. The assumptions made require that the distribution of pairs (Xi,Xj) decompose as the sum of an absolutely continuous measure with another measure concentrated on the diagonal of IR ~ IR satisfying a further absolute continuity with respect to the Lebesgue measure on this diagonal. For the convergence in probability we find the usual convergence rate on the bandwidth, whereas for the almost sure convergence we need to require that the bandwidth does not decrease to fast and that the kernel is of bounded variation. This assumption on the kernel is also required for the asymptotic normality, together with a slightly strengthened version of the usual decrease rate on the bandwidth. The assumption of bounded variation on the kernel is needed as a consequence of the dependence structure we are dealing with, as association is only preserved by monotone transformations.