Bivariante distribution function estimation for associated variables

Abstract

The estimation of distribution functions of pairs of associated variables is addressed based on a kernel estimator. This problem is motivated by the need to approximate covariance functions appearing as the limiting covariances of the empirical process sequence. Results characterizing the asymptotics and convergence rates of the estimator are obtained. From these we derive the optimal bandwidth convergence rate, which is of order n-1 . Finally, we give conditions for the asymptotic normality of the finite dimensional distributions, characterizing their limit covariance matrix. Besides some usual conditions on the kernel function, the conditions typically impose a convenient decrease rate on the covariances Cov(X1 , X n ).