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Zbl 0746.62056
Bingham, Christopher; Chang, Ted; Richards, Donald
Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. (English)
[J] J. Multivariate Anal. 41, No.2, 314-337 (1992). ISSN 0047-259X

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 $\exp(\hbox{Tr }GX\sp t\sb 0X)$. Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to $\exp(x\sp tGx)$, on the unit sphere $S\sp{p-1}$ 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 $S\sp{p- 1}$ 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.\par 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 2000:: *62H10 Multivariate distributions of statistics
62E20 Asymptotic distribution theory in statistics
33C90 Appl. of hypergeometric functions
62A01 Foundational and philosophical topics
86A60 Geological problems
62P99 Appl. of statistics

Keywords: estimated rotations; asymptotic expansions; tectonic plate reconstructions; beef carcasses; zonal polynomials; hypergeometric functions of matrix argument; approximations; matrix Fisher distributions; $SO(p)$; $O(p)$; Haar measure; Bingham distribution; unit sphere; Euclidean $p$-dimensional space; exact conditional distribution; maximum likelihood estimate; spherical regression model; Procrustes analysis; confidence region; unknown rotation

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