PEARSON TYPE VII DISTRIBUTIONS ON SPHERES

Abstract

ABSTRACT It is known that von Mises-Fisher or Langevin distributions have some analogous properties to multivariate normal distributions and are generated through conditioning suitable multivariate normal distributions. In this paper a new distribution on the unit sphere of arbitrary dimension, which is called the Pearson type VII distribution on the unit sphere, is obtained by conditioning scale mixtures of normal distributions with gamma weight and some properties of the proposed distribution are studied. A special case of the Pearson type VII disitribution on the unit sphere, which is called the t-distribution with n degrees of freedom on the unit sphere in this paper, converges to the von Mises-Fisher distribution as n tends to infinity. The situation is analogous to that of multivariate Student t- and normal distributions in Euclidean space. A new measure of similarity on spheres is also proposed.