Loss and risk in smoothing parameter selection

For several years there has been debate over the relative merits of loss and risk as measures of the performance of nonparametric density estimators. In the way that this debate has dealt with risk, it has largely ignored the fact that any practical bandwidth selection rule must produce a random bandwidth. Existing theory for risk of density estimators is almost invariably concerned with nonrandom bandwidths. In the present paper we examine two different definitions of risk, both of them appropriate to circumstances where the bandwidth is random. Arguments in favor of, and motivations for, each approach are presented, including formulation of appropriate decision-theoretic frameworks. It is shown that the two approaches can give diametrically opposite answers to the question of which of two competing bandwidths selection rules is superior. Technical results include some surprising Conclusions about the nonexistence of risks, and even of moments of some common data-driven bandwidths under the usual assumptions.