Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data

Authors:
Tatsuya Kubokawa;Muni S. Srivastava
Affiliations:
Faculty of Economics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan;Department of Statistics, University of Toronto, 100 St George Street, Toronto, Ontario, Canada M5S 3G3
Venue:
Journal of Multivariate Analysis
Year:
2008

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Abstract

In this article, the Stein-Haff identity is established for a singular Wishart distribution with a positive definite mean matrix but with the dimension larger than the degrees of freedom. This identity is then used to obtain estimators of the precision matrix improving on the estimator based on the Moore-Penrose inverse of the Wishart matrix under the Efron-Morris loss function and its variants. Ridge-type empirical Bayes estimators of the precision matrix are also given and their dominance properties over the usual one are shown using this identity. Finally, these precision estimators are used in a quadratic discriminant rule, and it is shown through simulation that discriminant methods based on the ridge-type empirical Bayes estimators provide higher correct classification rates.