Inadmissibility of the Maximum Likekihood Estimator of Normal Covariance Matrices with the Lattice Conditional Independence

Lattice conditional independence (LCI) models introduced by S. A. Andersson and M. D. Perlman (1993, Ann. Statist.21, 1318–1358) have the pleasant feature of admitting explicit maximum likelihood estimators and likelihood ratio test statistics. This is because the likelihood function and parameter space for a LCI model can be factored into products of conditional likelihood functions and parameter spaces, where the standard multivariate techniques can be applied. In this paper we consider the problem of estimating the covariance matrices under LCI restriction in a decision theoretic setup. The Stein loss function is used in this study and, using the factorization mentioned above, minimax estimators are obtained. Since the maximum likelihood estimator has constant risk and is different from the minimax estimator, this shows that the maximum likelihood estimator under LCI restriction inadmissible. These results extend those obtained by W. James and C. Stein (1960, in “Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics, and Probability,” Vol. 1, pp. 360–380, Univ. of California Press, Berkeley, CA) and D. K. Dey and C. Srinivasan (1985, Ann. Statist.13, 1581–1591) for estimating normal covariance matrices to the LCI models.