Improved Nonnegative Estimation of Variance Components in Balanced Multivariate Mixed Models

T Mathew, A. Niyogi and B. K. Sinha

Journal of Multivariate Analysis, 1994, vol. 51, issue 1, 83-101

Abstract: Consider the independent Wishart matrices S1 W([Sigma] + [lambda][Theta],q1) and S2 W([Sigma], q2) where [Sigma] is an unknown positive definite (p.d.) matrix, [Theta] is an unknown nonnegative definite (n.n.d.) matrix, and [lambda] is a known positive scalar. For the estimation of [Theta], a class of estimators of the form [Theta](c,[epsilon]) = (c/[lambda]){S1/q1 - [epsilon](S2/q2)} (c >= 0, [epsilon] 0, [epsilon] > 0, the estimator obtained by taking the positive part of [Theta](c, [epsilon]) results in an n.n.d. estimator, say [Theta](c, [epsilon]) +, that is uniformly better than [Theta]U. Numerical results indicate that in terms of mean squared error, [Theta](c, [epsilon]) + performs much better than both [Theta]U and the restricted maximum likelihood estimator [Theta]REML of [Theta]. Similar results are also obtained for the nonnegative estimation of tr [Theta] and a'[Theta]a, where a is an arbitrary nonzero vector. For estimating [Sigma], we have derived estimators that are claimed to be uniformly better than the unbiased estimator [Sigma]U = S2/q2 under the squared error loss function and the entropy loss function. We have been able to establish the claim only in the bivariate case. Numerical results are reported showing the risk improvement of our proposed estimators of [Sigma].

Date: 1994
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