Unconstrained Pilot Selectors for Smoothed Cross-Validation

Two of the most useful multivariate bandwidth selection techniques are the plug-in and cross-validation methods. The smoothed version of the cross-validation method is known to reduce the variability of its non-smoothed counterpart; however, it shares with the plug-in choice the need for a pilot bandwidth matrix. Owing to the mathematical difficulties encountered in the optimal pilot choice, it is common to restrict this pilot matrix to be a scalar multiple of the identity matrix, at the expense of losing the flexibility afforded by the unconstrained approach. Here we show how to overcome these difficulties and propose a smoothed cross-validation selector using an unconstrained pilot matrix. Our numerical results indicate that the unconstrained selector outperforms the constrained one in practice, and is a viable competitor to unconstrained plug-in selectors.