IMPROVED NONNEGATIVE ESTIMATION OF VARIANCE-COMPONENTS IN BALANCED MULTIVARIATE MIXED MODELS

Consider the independent Wishart matrices S-1 similar to W(Sigma + lambda Theta,q(1)) and S2 similar to W(Sigma, q(2)) where Sigma is an unknown positive definite (p.d.) matrix, Theta is an unknown nonnegative definite (n.n.d.) matrix, and 1 is a known positive scalar. For the estimation of Theta, a class of estimators of the form ($) over cap Theta((c,epsilon)) = (c/lambda){S-1/q(1) - epsilon(S-2/q(2))} (c greater than or equal to 0, epsilon less than or equal to 1), uniformly better than the unbiased estimator ($) over cap Theta(U) = (1/lambda){S-1/q(1) - S-2/q(2)}, is derived (for the squared error loss function). Necessary and sufficient conditions are obtained For the existence of an n.n.d. estimator of the form ($)over cap>Theta((c,epsilon)) uniformly better than ($) over cap Theta(U). It turns out that such an n.n.d. estimator exists only under restrictive conditions. However, for a suitable choice of c > 0, epsilon > 0, the estimator obtained by taking the positive part of ($) over cap Theta((c,epsilon)) results in an n.n.d. estimator, say ($) over cap Theta((c,epsilon)+), that is uniformly better than ($) over cap Theta(U). Numerical results indicate that in terms of mean squared error, ($) over cap Theta((c,epsilon)+) performs much better than both ($) over cap Theta(U) and the restricted maximum likelihood estimator ($) over cap 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 ($) over cap Sigma(U) = S-2/q(2) 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. (C) 1994 Academic Press, Inc.