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Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and procrustes analysis

Author

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  • Bingham, Christopher
  • Chang, Ted
  • Richards, Donald

Abstract

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.

Suggested Citation

  • Bingham, Christopher & Chang, Ted & Richards, Donald, 1992. "Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and procrustes analysis," Journal of Multivariate Analysis, Elsevier, vol. 41(2), pages 314-337, May.
  • Handle: RePEc:eee:jmvana:v:41:y:1992:i:2:p:314-337
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    Cited by:

    1. W. V. Félix de Lima & A. D. C. Nascimento & G. J. A. Amaral, 2021. "Entropy-based pivotal statistics for multi-sample problems in planar shape," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 153-178, March.
    2. Gao, Hongsheng & Smith, Peter J., 2000. "A Determinant Representation for the Distribution of Quadratic Forms in Complex Normal Vectors," Journal of Multivariate Analysis, Elsevier, vol. 73(2), pages 155-165, May.
    3. Valdevino Félix de Lima, Wenia & David Costa do Nascimento, Abraão & José Amorim do Amaral, Getúlio, 2021. "Distance-based tests for planar shape," Journal of Multivariate Analysis, Elsevier, vol. 184(C).

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