An Antipodally Symmetric Distribution on the Sphere

Abstract

The distribution Ψ(x;Z,M)=const.exp(tr(ZMTxxTM))Ψ(x;Z,M)=const.exp⁡(tr(ZMTxxTM))\Psi(\mathbf{x}; Z, M) = \operatorname{const}. \exp(\mathrm{tr} (ZM^T \mathbf{xx}^T M)) on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix ZZZ and the orthogonal orientation matrix M.ΨM.ΨM. \Psi is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for ZZZ and MMM are found and their asymptotic properties elucidated. Inference procedures, including tests for isotropy and circular symmetry, are proposed. The application of ΨΨ\Psi is illustrated by a numerical example.