Large Sample Optimality of Least Squares Cross-Validation in Density Estimation

Abstract

We prove that the method of cross-validation suggested by A. W. Bowman and M. Rudemo achieves its goal of minimising integrated square error, in an asymptotic sense. The tail conditions we impose are o~ly slightly more severe than the hypothesis of finite variance, and so least squares cross-validation does not exhibit the pathological behaviour which has been observed for Kullback-Leibler cross-validation. This is apparently the first time that a cross-validatory procedure for density estimation has been shown to be asymptotically optimal, rather then simply consistent.