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, , of its exact mean integrated squared error (MISE) and show that it is asymptotically equivalent to the bandwidth that minimizes the leading terms of the MISE, together with the order of convergence of the relative error . 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.