On the existence and limit behavior of the optimal bandwidth for kernel density estimation

Abstract

We prove, under mild conditions, the existence of a minimizer of the exact mean integrated square error of a kernel density estimator as a function of the bandwidth. When it exists, we also show some expected limit properties of this optimal bandwidth-, in fact, for two common situations, Theorem 3 gives the exact value for the limit. Surprisingly, in some special cases (when using superkernels or the sine kernel for estimating some classes of densities), this limit is strictly positive and a global zero-bias bandwidth can be chosen.