On the Information Matrix for Symmetric Distributions on the Hypersphere

Abstract

A general definition of symmetry for a directional model is given. Then the information matrix for the parameters indexing a symmetric density is shown to be block-diagonal: one block for the location and one block for the shape. This result is used to construct an algorithm for the efficient estimation of the parameters. Examples are given; a new symmetric distribution, the $FB_6$ is introduced; it generalizes the classical distributions on the hypersphere.