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

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

Let 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 for fixed 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 . Besides of some usual conditions on the kernel function, the conditions typically impose a convenient increase rate on the covariances .