On a family of two–piece circular distributions

Abstract

A new way of constructing flexible and unimodal circular models, focusing on the modal direction, is proposed. Starting from a base symmetric density and a weight function, a two–piece four parameters density is introduced. The proposed density provides an extension of the base density to allow for sharply peaked and flat–topped unimodal distributions as well as a wide range of skewness. In particular, it generalizes some well–known peakedness–free models such as the Batschelet and Papakonstantinou densities. The four parameters of the model have a clear interpretation: modal direction, concentration, peakedness at the left and at the right of the modal direction. Symmetric submodels are obtained when the peakedness parameters are equal. The main properties related to the shape of the new density are presented and asymptotic results for maximum likelihood estimators are derived. An illustrative application concerning the flight orientation of migrating raptors is investigated.