© 2002 by Biometrika Trust
Spectral models for covariance matrices
1 Department of Mathematical Sciences, Montana State University, Bozeman, Montana 59717-2400, U.S.Arjboik{at}math.montana.edu
A new model for the simultaneous eigenstructure of multiple covariance matrices is proposed.The model is much more flexible than existing models and subsumes most of them as special cases. A Fisher scoring algorithm for computing maximum likelihood estimates of the parameters under normality is given. Asymptotic distributions of the estimators are derived under normality as well as under arbitrary distributions having finite fourth-order cumulants. Special attention is given to elliptically contoured distributions. Likelihood ratio tests are described and sufficient conditions are given under which the test statistics are asymptotically distributed as chi-squared random variables. Procedures are derived for evaluating Bartlett corrections under normality. Some conjectures made by Flury (1988) are verified; others are refuted. A small simulation study of the adequacy of the Bartlett correction is described and the new procedures are illustrated on two datasets.
Key Words: Bartlett correction; Common principal components; Common space; Eigenprojection; Eigenspace; Elliptically contoured distribution; Maximum likelihood; Orthogonal matrix; Partial sphericity; Principal components; Proportional covariance matrices; Sphericity
Received October 2000. Revised May 2001
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