Bootstrap bandwidth selection using an $h$-dependent pilot bandwidth

Abstract

The problem of choosing the bandwidth h for kernel density estimation is considered. All the plug-in-type bandwidth selection methods require the use of a pilot bandwidth g. The usual way to make an h-dependent choice of g is by obtaining their asymptotic expressions separately and solving the two equations. In contrast, we obtain the asymptotically optimal value of g for every fixed h, thus making our selection `less asymptotic’. Exact error expressions show that some usually assumed hypotheses have to be discarded in the asymptotic study in this case. Two versions of a new bandwidth selector based on this idea are proposed, and their properties are analysed through theoretical results and a simulation study.