McMASTER UNIVERSITY STATISTICS SEMINAR

Week of January 24 - 28, 2000

SPEAKER:	Dr Maung Min-Oo Department of Mathematics & Statistics, McMaster University
TITLE:	"Geometry of multivariate normal distributions"
DAY:	Wednesday, January 26, 2000
TIME:	3:30 p.m. [Coffee & cookies in BSB-202 at 3:00 p.m.]
PLACE:	BSB-108

SUMMARY

Geometric methods have been used by a number of statisticians in parametric inference problems. The Fisher information matrix defines a Riemannian metric on the parameter space and geometric (i.e. coordinate independent) properties of the model can be studied. The two main models used are exponential families and transformational families with symmetries. In this talk we will concentrate on transformational models. After a brief but general introduction to some of the differential-geometric ideas used in parametric estimation, we will discuss the following topics:

1. A new maximally symmetric Riemannian metric for the space of all multivariate normal distributions. This metric coincides with the Fisher metric in dimension 1, but is slightly different in higher dimensions. (Joint work with M. Lovric and E. Ruh).
2. The convergence of the discrete linear Kalman filter. This can be proved using a contraction map on the space of all symmetric positive definite matrices. The choice of the metric is significant in controlling the rate of convergence. (This is based on work by P. Bougerol.)

If time permits, we will also give a short account of random matrix theory and its relation to the Wishart distribution.

One goal of this talk is to show that there is a link between differential geometry and certain aspects of multivariate statistics.

ABOUT THE SPEAKER

Dr Maung Min-Oo obtained his B.Sc. from Rangoon University in Burma and his Dip. Math. and Dr. rer. nat. from Bonn University in Germany. He is Professor in the Department of Mathematics & Statistics at McMaster University. His interests include Differential Geometry, Geometric Analysis, General Relativity, and Financial Mathematics.

REFERENCES

The references below have been suggested by Dr Min-Oo as useful background for his talk. Papers [1] and [2] have been placed on reserve at Thode Library (STATS 770: Statistics Seminar).

Papers:

[1] M.Lovric, M.Min-Oo and E.Ruh: "Multivariate normal distributions parametrized as a Riemannian symmetric space"; preprint, McMaster University, 1998. (submitted to J. Multivariate Analysis).

[2] P.Bougerol: "Kalman filtering with random coefficients and contractions"; SIAM J. Control and Optimization vol.31.4, pp. 942-959, 1993.

[3] M.Calvo and J.M.Oller: "A distance between multivariate normal distributions based on an embedding into the Siegel group", J. Multivariate Analysis vol.35, pp. 223-242, 1990.

[4] L.T.Skovgaard: "A Riemannian geometry of the multivariate normal model"; Scand. J. Statistics vol.11, pp. 211-223, 1984.

Books:

[1] O.E.Barndorff-Nielsen: Parametric statistical models and likelihood, Lecture Notes in Statistics #50, Springer-Verlag, 1988.

[2] M.K.Murray and J.W.Rice: Differential Geometry and Statistics, Monographs on Statistics and Applied Probability, Chapman and Hall, 1993.

[3] A.V.Balakrishnan: Kalman filtering theory, Optimization Software Inc., 1984.