Do-Validation for Kernel Density Estimation

Abstract

Bandwidth selection in kernel density estimation is one of the fundamental model selection problems of mathematical statistics. The study of this problem took major steps forward with the articles of Hall and Marron (1987) and Hall and Johnstone (1992). Since then, the focus seems to have been on various versions of implementing the so-called plug-in method aimed at estimating the minimum mean integrated squared error (MISE). The most successful of these efforts still seems to be the plug-in method of Sheather and Jones (1991) or Park and Marron (1990) that we also use as a benchmark in this article. In this article we derive a new theorem deriving the asymptotic theory for linear combinations of bandwidths obtained from different selectors as, for example, direct and indirect cross-validation and plug-in, where we take advantage of recent advances in the study of indirect cross-validation; see Hart and Yi (1998), Hart and Lee (2005), and Savchuk, Hart, and Sheather (2008, 2010). We conclude that the slow convergence of data-driven bandwidths implies that once asymptotic theory is close to that of the plug-in, then it is the practical implementation that counts. This insight led us to a bandwidth selector search with the symmetrized version of one-sided cross-validation as a clear winner.