Estimation of a Multivariate Normal Covariance Matrix with Staircase Pattern Data

• Activity Number: 351
• Type: Topic Contributed
• Date/Time: Wednesday, August 10, 2005 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Xiaoqian Sun*+ and Dongchu Sun
• Companies: University of Missouri, Columbia and University of Missouri, Columbia
• Keywords: staircase pattern data ; covariance matrix ; Bartlett decomposition ; Bartlett decomposition ; Jeffreys prior ; reference prior

Abstract

In this paper, we study the problem of estimating a multivariate normal covariance matrix with staircase pattern data. Two kinds of parameterizations in terms of the covariance matrix are used. One is Cholesky decomposition and the other is Bartlett decomposition. Based on Cholesky decomposition of the covariance matrix, the closed form of the maximum likelihood estimate of the covariance matrix is given. Using Bayesian method, we prove the best equivariant estimate of the covariance matrix with respect to the special group related to Cholesky decomposition uniquely exists under the Stein loss. Consequently, the MLE of the covariance matrix is inadmissible under the Stein loss. Our method also can be applied to other invariant loss functions such as the entropy and symmetric loss. In addition, based on Bartlett decomposition of the covariance matrix, the Jeffreys prior and reference prior of the covariance matrix with staircase pattern data also are obtained. Our reference prior is different from Berger and Yang's reference prior.