Transformation of Circular Random Variables Based on Circular Distribution Functions

Abstract

In this paper, we propose a new method of generating new circular densities based on the product of two base circular densities, which we shall call circular product method. Very general results are provided for the properties of the proposed families of circular product models, including the trigonometric moments, their maximum of entropy, random variate generation, their finite mixture and modality properties. In particular, we focus our attention on a subfamily of the general family with one of the periodic densities as the cardioid density. The modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for this special case which employs the von Mises density as the other periodic density function. Indeed, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of guncrimes and also a simulated data set.