Asymptotic behaviour of multistage plug-in bandwidth selections for kernel distribution function estimators

Abstract

Given X 1, …, X n independent real random variables with common but unknown absolutely continuous distribution function F, we study the asymptotic behaviour of two classes of multistage plug-in bandwidth selectors for the kernel distribution function estimator F¯ n , on the basis of two asymptotic approximations of the optimal bandwidth h MISE that minimizes the mean integrated square error E∫{F¯ n (x)−F(x)}2 dx. The second asymptotic approximation we consider is, to our knowledge, new in the literature. Although a better rate of convergence for h MISE could be obtained by a multistage plug-in procedure based on this new asymptotic approximation, we prove that, from an asymptotic point of view, there is not a substantial difference between the two classes of associated kernel distribution function estimators in the sense of the integrated square error. For finite sample sizes, a simulation study indicates that the plug-in methods based on the new asymptotic approximation of the optimal bandwidth are superior to the corresponding one based on the asymptotic approximation usually considered in the literature. Some comparisons with the cross-validation procedure proposed by Bowman et al. [Bowman, A., Hall, P. and Prvan, T., 1998, Bandwidth selection for the smoothing of distribution functions. Biometrika, 85, 799–808.] are also presented.