Sankhya: The Indian Journal of Statistics

1996, Volume 58, Series A, Pt. 1, pp. 101--114

POLAR COORDINATES IN Rnp; APPLICATION TO THE COMPUTATION OF THE WISHART AND BETA LAWS

By

A. CADET, INSEE, France

SUMMARY.  During some time the multivariate statistical analysis has been limited to the study of the normal laws and of its derived laws. Then it was realised that some properties of the normal laws remained unchanged for more general distributions, the spherical and elliptical laws, the most spectacular result being that the ratio F follows the same law  beta for all those distributions. For that demonstrations rely upon changing in polar coordinates in Rn which become a central tool. Later some author tried to generalise these results to matrices without getting the same tool. In this article we develop a theory of polar coordinates for matrices while showing the application in statistics (computaton of the Wishart and matrix beta laws ). The results use the differential and integral calculus of manifolds, theory which is not yet fully known among statisticians. A state of its present use of elliptical distributions can be found in: Brandorff-Nielsen, Blaesild, Eriksen (1989), Muirhead (1982), Fang, Kotz, Ng(1990), Fang, Zhang (1990), and also my article(Cadet (1991)).

AMS (1980) subject classification.  62H10, 28A75, 28C10, 51A50, 58C35.

Key words and phrases. Polar coordinates, multivariate statistics, Wishart, beta, orthogonal group, elliptical distributions.

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