Almost Sure Convergence of Kernel Bivariate Distribution Function Estimator under Negative Association

Let {Xn ,n=>1} be a strictly stationary sequence of negatively associated random variables, with common distribution function F. In this paper, we consider the estimation of the two-dimensional distribution function of (X1, Xk+1) for fixed $K /in N$ based on kernel type estimators. We introduce asymptotic normality and properties and moments. From these we derive the optimal bandwidth convergence rate, which is of order n-1. Besides of some usual conditions on the kernel function, the conditions typically impose a convenient increase rate on the covariances cov(X1,Xn).