Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation

Abstract

Let ho, ĥo and ĥc be the windows which minimise mean integrated square error, integrated square error and the least-squares cross-validatory criterion, respectively, for kernel density estimates. It is argued that ĥo, not ho, should be the benchmark for comparing different data-driven approaches to the determination of window size. Asymptotic properties of ho-ĥo and ĥc-ĥo, and of differences between integrated square errors evaluated at these windows, are derived. It is shown that in comparison to the benchmark ĥo, the observable window ĥc performs as well as the so-called "optimal" but unattainable window ho, to both first and second order. © 1987 Springer-Verlag.