TY - JOUR

T1 - Approximating the matrix Fisher and Bingham distributions

T2 - Applications to spherical regression and procrustes analysis

AU - Bingham, Christopher

AU - Chang, Ted

AU - Richards, Donald

N1 - Funding Information:
Received November 16, 1990; revised August 29, 1991. AMS 1980 subject classification: primary 62H10, 62E20, 86A60. Key words and phrases: Bingham distribution, matrix Fisher distribution, spherical regression, where’s the beef, estimated rotations. Procrustes analysis, asymptotic expansions, tectonic plate reconstructions, beef carcasses, zonal polynomials, hypergeometric functions of matrix argument. * The second named author wishes to acknowledge support from the Kansas Geological Survey and NSF Grant DMS-8901494. The third named author was supported by NSF Grant DMS-8802929.

PY - 1992/5

Y1 - 1992/5

N2 - 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(Tr GX0tX). Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to exp(xtGx), on the unit sphere Sp-1 in Euclidean p-dimensional space. The matrix Fisher distribution arises as the exact conditional distribution of the maximum likelihood estmate of the unknown orthogonal matrix in the spherical regression model on Sp-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.

AB - 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(Tr GX0tX). Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to exp(xtGx), on the unit sphere Sp-1 in Euclidean p-dimensional space. The matrix Fisher distribution arises as the exact conditional distribution of the maximum likelihood estmate of the unknown orthogonal matrix in the spherical regression model on Sp-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.

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U2 - 10.1016/0047-259X(92)90072-N

DO - 10.1016/0047-259X(92)90072-N

M3 - Article

AN - SCOPUS:38249012897

VL - 41

SP - 314

EP - 337

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -