A Parzen-Rosenblatt type density estimator for circular data: exact and asymptotic optimal bandwidths

Abstract

For the Parzen-Rosenblatt type density estimator for circular data we prove the existence of a minimizer, h_{MISE}(f;K,n), of its exact mean integrated squared error (MISE) and show that it is asymptotically equivalent to the bandwidth h_{AMISE}(f;K,n) that minimizes the leading terms of the MISE, together with the order of convergence of the relative error h_{AMISE}(f;K,n)/h_{MISE}(f;K,n)-1. Some small and moderate sample size comparisons between the two bandwidths are also presented when the underlying density is a mixture of von Mises densities.