Estimation of covariance matrices in fixed and mixed effects linear models

Abstract

The estimation of the covariance matrix or the multivariate components of variance is considered in the multivariate linear regression models with effects being fixed or random. In this paper, we propose a new method to show that usual unbiased estimators are improved on by the truncated estimators. The method is based on the Stein-Haff identity, namely the integration by parts in the Wishart distribution, and it allows us to handle the general types of scale-equivariant estimators as well as the general fixed or mixed effects linear models.

Better Estimates of Genetic Covariance Matrices by “Bending” Using Penalized Maximum Likelihood

“…Loh (1991), Mathew et al (1994), Srivastava and Kubokawa (1999), and Kubokawa and Tsai (2006) considered estimation for two independent Wishart matrices, such as B and W in the one-way classification. Minimizing the sum of entropy losses, they derived different types of joint estimators, analogous to those proposed for a single matrix, and showed that improved estimators were available that had lower risk than the unbiased or REML estimators.…”

Estimation of Wishart mean matrices under simple tree ordering

“…. , p. When k = 2, Kubokawa and Tsai [8] showed that the estimator S i dominates the James-Stein-type minimax estimator JS i under the Kullback-Leibler loss function for each i, i = 1, 2. Let S = ( S 1 , .…”

“…Kubokawa and Tsai [8] further studied the estimation of multivariate components of variance in the multivariate linear regression models with effects being mixed via the Stein-Haff Wishart identity and showed that each MLE dominates the corresponding moment estimator (unrestricted MLE), but there is no report on simultaneous study. In practice, we may be interested in the estimation problem of unknown covariance matrices for the completely balanced multivariate multi-way random effects models without interactions, which allow more than two unknown covariance matrices to be involved.…”

“…However, it is difficult to apply their method to general types of estimators in more complicated models. To overcome it, Kubokawa and Tsai [8] adopted the Stein-Haff identity to study risk dominances for the one-way multivariate mixed effects models (namely, for the case of only two unknown matrices involved). In this paper, we further extend the results of Kubokawa and Tsai [8] to the cases when more than two matrices (k 3) are involved.…”

“…= 9 n =(15,11, 20) n = (33, 70, 48) a = (1, 3, 1, 5, 1, 7, 1, 9, 1) a = (1, 1, 1, 1, 1, 1, 1, 1, 1) b =(1,4,3,6,5,8,7,10,9) b = (211, 87, 92, 1, 67, 33, 2, 2001, 30) c = (3, 12, 13, 14, 2, 16, 12, 18, 2) c = (90, 3050, 85, 2, 3, 12, 945, 12, 387)tree ordering set K, but fromTable 1we observe that the risks of them are not always less than those of the restricted MLE.…”