How much do plug-in bandwidth selectors adapt to non-smoothness?

Abstract

The asymptotic behaviour of some popular plug-in bandwidth selection procedures for kernel density estimators is reviewed for situations where the observations are drawn from a density which possibly has discontinuities or discontinuities in its derivatives. It is shown that the procedures, with the exception of the Gaussian reference bandwidth, adapt to some extent to non-smoothness. The adaptation results in automatic selection of a smaller bandwidth if the underlying density is less smooth. The amount of adaptation for different procedures is investigated. The considered bandwidth selection procedures do not fully adapt to the optimal bandwidths for non-smooth densities Special attention is paid to the bandwidth selector introduced by Sheather and Jones (1991) which is one of the most used procedures due to its good performance in finite samples.