×

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 \(\exp(\hbox{Tr }GX^ t_ 0X)\). Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to \(\exp(x^ tGx)\), 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 distribution 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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, G. A., An asymptotic expansion for the noncentral Wishart distribution, Ann. Math. Statist., 41, 1700-1707 (1970) · Zbl 0223.62066
[2] Barndorff-Nielsen, O.; Blaesild, P.; Jensen, J. L.; Jorgensen, B., Exponential transformation models, (Proc. Roy. Soc. London Ser. A, 379 (1982)), 41-65 · Zbl 0478.62005
[3] Bingham, C., An antipodally symmetric distribution on the sphere, Ann. Statist., 2, 1201-1225 (1974) · Zbl 0297.62010
[4] Bingham, C., (Expansions Related to a Hypergeometric Function Arising in an Antipodally Symmetric Distribution on the Sphere (1977), University of Leeds), Directional Data Analysis Project: Research Report 11
[5] Bingham, C.; Chang, T., (The Use of the Bingham Distribution in Spherical Regression Inference (1989), University of Minnesota, School of Statistics), Technical Report No. 519
[6] Chang, T., Spherical regression, Ann. Statist., 14, 907-924 (1986) · Zbl 0605.62079
[7] Chang, T., Spherical regression and the statistics of tectonic plate reconstructions, Internat. Statist. Rev. (1991), to appear
[8] Goodall, C., Procrustes methods in the statistical analysis of shape, J. Roy. Statist. Soc. Ser. B, 53, 285-339 (1991) · Zbl 0800.62346
[9] Gower, J. C., Generalized Procrustes analysis, Psychometrika, 40, 33-51 (1975) · Zbl 0305.62038
[10] Gross, K. I.; Richards, D. St. P., Hypergeometric functions on complex matrix space, Bull. Amer. Math. Soc. (N.S.), 24, 349-355 (1991) · Zbl 0731.33015
[11] Hanna, M.; Chang, T., On graphically representing the confidence region for an unknown rotation in three dimensions, Comput. & Geosc., 16, 163-194 (1990)
[12] James, A. T., Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist., 35, 475-501 (1964) · Zbl 0121.36605
[13] Kent, J., Asymptotic expansions for the Bingham distribution, Appl. Statist., 36, 139-144 (1987) · Zbl 0617.62052
[14] Kent, J., The complex Bingham distribution and shape distributions, (Summary of talk at Wilks Symposium Princeton University. Summary of talk at Wilks Symposium Princeton University, May 1990 (1990)) · Zbl 0806.62040
[15] Khatri, C. G.; Mardia, K. V., The von Mises-Fisher matrix distribution in orientation statistics, J. Roy. Statist. Soc. Ser. B, 39, 95-106 (1977) · Zbl 0356.62044
[16] Lehmann, E. L., (Testing Statistical Hypotheses (1986), Wiley: Wiley New York)
[17] Luke, Y. L., (The Special Functions and Their Approximations, Vol. 1 (1969), Academic Press: Academic Press New York)
[18] MacKenzie, J. K., The estimation of an orientation relationship, Acta Crystallogr., 10, 61-62 (1957)
[19] Muirhead, R. J., Latent roots and matrix variates: A review of some asymptotic results, Ann. Statist., 6, 5-33 (1978) · Zbl 0375.62050
[20] Muirhead, R. J., (Aspects of Multivariate Statistical Theory (1982), Wiley: Wiley New York) · Zbl 0556.62028
[21] Prentice, M. J., Orientation statistics without parametric assumptions, J. Roy. Statist. Soc. Ser. B, 48, 214-222 (1986) · Zbl 0601.62071
[22] Rivest, L., Spherical regression for concentrated Fisher-von Mises distributions, Ann. Statist., 17, 307-317 (1989) · Zbl 0669.62041
[23] Stephens, M. A., Vector correlation, Biometrika, 66, 41-48 (1979) · Zbl 0402.62033
[24] Sukhatme, P. V.; Sukhatme, B. V., (Sampling Theory of Surveys with Applications (1970), Iowa State University Press: Iowa State University Press Ames) · Zbl 0239.62008
[25] Wahba, G., Section on problems and solutions: A least squares estimate of satellite attitude, SIAM Rev., 8, 384-385 (1966)
[26] Watson, G. S., (Statistics on Spheres (1983), Wiley: Wiley New York) · Zbl 0646.62045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.