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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 $SO\left(p\right)$ and on $O\left(p\right)$ whose density with respect to Haar measure is proportional to $exp\left(\text{Tr}\phantom{\rule{4.pt}{0ex}}G{X}_{0}^{t}X\right)$. Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to $exp\left({x}^{t}Gx\right)$, 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:
 62H10 Multivariate distributions of statistics 62E20 Asymptotic distribution theory in statistics 33C90 Applications of hypergeometric functions 62A01 Foundations and philosophical topics in statistics 86A60 Geological problems 62P99 Applications of statistics