An Antipodally Symmetric Distribution on the Sphere

The distribution Ψ(x; Z, M) = const. exp(tr (ZMT xxT M)) on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix Z and the orthogonal orientation matrix M. Ψ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for Z and M are found and their asymptotic properties elucidated. Inference procedures, including tests for isotropy and circular symmetry, are proposed. The application of Ψ is illustrated by a numerical example.