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.