ON THE EXACT AND THE APPROXIMATE MEAN INTEGRATED SQUARE ERROR FOR THE KERNEL DISTRIBUTION FUNCTION ESTIMATOR

Abstract

The asymptotic mean integrated square error (AMISE) is used as an approximate measure of error for the mean integrated square error (MISE). The exact MISE for kernel density estimator is discussed by Marron and Wand [1]. In this investigation we discuss the exact MISE and the AMISE for the cumulative distribution function estimate. Also, we compare between the optimal bandwidth that minimize the AMISE and that minimize the MISE. In addition, through simulations these optimal bandwidths are compared with the bandwidth selectors using the least square cross - validation (LSCV), biased cross - validation (BCV), and direct plug -in (DPI) techniques.