A Cross-Validation Bandwidth Choice for Kernel Density Estimates with Selection Biased Data

This paper studies the risks and bandwidth choices of a kernel estimate of the underlying density when the data are obtained fromsindependent biased samples. The main results of this paper give the asymptotic representation of the integrated squared errors and the mean integrated squared errors of the estimate and establish a cross-validation criterion for bandwidth selection. This kernel density estimate is shown to be asymptotically superior to many other intuitive kernel density estimates. The data-driven cross-validation bandwidth is shown to be asymptotically optimal in the sense of Stone (1984,Ann. Statist.12, 1285–1297). The finite sample properties of the cross-validation bandwidth are investigated through a Monte Carlo simulation.