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Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. (English) Zbl 0746.62056

Summary: We obtain approximations to the distribution of the exponent in the matrix Fisher distributions on SO(p) and on O(p) whose density with respect to Haar measure is proportional to exp(TrGX 0 t X). Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to exp(x t Gx), on the unit sphere S p-1 in Euclidean p- dimensional space. The matrix Fisher distribution arises as the exact conditional distribution of the maximum likelihood estimate of the unknown orthogonal matrix in the spherical regression model on S p-1 with Fisher distributed errors. It also arises as the exact conditional distribution of the maximum likelihood estimate of the unknown orthogonal matrix in a model of Procrustes analysis in which location and orientation, but not scale, changes are allowed.

These methods allow determination of a confidence region for the unknown rotation for moderate sample sizes with moderate error concentrations when the error concentration parameter is known.

MSC:
62H10Multivariate distributions of statistics
62E20Asymptotic distribution theory in statistics
33C90Applications of hypergeometric functions
62A01Foundations and philosophical topics in statistics
86A60Geological problems
62P99Applications of statistics