Exponential family of circular distributions
Sung‐Su Kim · Journal of the Korean Data and Information Science Society · 2011
Abstract In this paper, we show that any circular density can be closely approximated byan exponential family of distributions. Therefore we propose an exponential familyof distributions as a new family of circular distributions, which is absolutely suitableto model any shape of circular distributions. In this family of circular distributions,the trigonometric moments are found to be the uniformly minimum variance unbiasedestimators (UMVUEs) of the parameters of distribution. Simulation result and good-ness of t test using an asymmetric real data set show usefulness of the novel circulardistribution.Keywords: Approximation, circular distribution, trigonometric polynomial, uniformlyminimum variance unbiased estimator. 1. Introduction Circular random variables are found in various areas of research such as biology, medicine,just to name a few. Because of the periodic nature of a circular variable, it is necessaryto use a circular distribution to model a circular variable. Up to date, there are so manycircular distributions available in literatures and books. Some of them are exible enoughto model asymmetric or multimodal circular distributions. For various types of circulardistribution, including von Mises (VM) or circular normal distribution, the readers can referto Jammalamadaka and SenGupta (2001).The circular normal distribution, which is symmetric, has been mainly used to model acircular random variable. However, circular distributions are rarely symmetric, i.e. they areusually asymmetric and even multi-modal. Therefore, the VM distribution is not suitableto model such a data set. In fact, this is also the case in linear statistical analysis (Arnoldand Beaver, 2000; Azzalini, 1985) that the normal distribution is often not suitable. Oneway to model an asymmetric and multimodal distribution is using a mixture of von Misesdistributions (Batschelet, 1981). Another model suitable for an aymmetric and/or multi-modal circular distribution is based on nonnegative trigonometric sums (Fernandez-Duran,2004). Other existing asymmetric and/or bimodal circular distributions are appeared inJammadamalaka and Kozubowski (2004), Getto and Jammalamadaka (2007), and Umbachand Jammalamadaka (2009).