The effect of discretization error on bandwidth selection for kernel density estimation

Abstract

It is well known that the cross-validation score function of discretized data often tends to -∞ as the bandwidth tends to zero. This disturbing property causes some difficulty in applying cross-validation to discretized data. Since all data are rounded to some degree, the phenomenon limits the application of cross-validation in practice. Based on characteristic functions, the cause of the difficulty is explained, and a simple modification is suggested. Under some conditions, it is shown that the bandwidth selected by the proposed procedure is asymptotically equivalent to the bandwidth selected by cross-validation of the original unrounded data. The advantages of the proposed procedure are demonstrated in two examples.