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

TLDR: This article showed that least square cross-validation is asymptotically optimal for density estimation, rather then simply consistent, in the sense that the tail conditions are only slightly more severe than the hypothesis of finite variance.

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 only 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.