Smooth Simultaneous Confidence Bands For Cumulative Distribution Functions Under Dependent Variables

Literatures have developed that a plug-in kernel estimator can be used as a smooth estimator of a cumulative distribution function with uniform convergence if independent samples given. Does this conclusion still hold in the case of dependent samples?In this paper, we propose an smooth estimator called kernel distribution estimator based on the dependent samples, and then prove the uniform convergence between this estimator and the empirical distribution function if some mild assumptions are satisfied. While according to the knowledge about mixing sequence in some relating paper,it has been established the uniform convergence between the cumulative distribution and the empirical distribution function under certain assumptions. So when the both condition are satisfied, we get the convergence between the plug-in kernel estimator and the cumulative distribution function. Furthermore, we construct a smooth confidence band and apply this method to an extensive simulation study. In this paper, we use two di?erent automatic bandwidths to make simulations and the simulation results confirm our theory.