Improved Nonnegative Estimation of Variance Components in Balanced Multivariate Mixed Models

Abstract

Consider the independent Wishart matrices S1W(Σ + λΘ,q1) and S2W(Σ, q2) where Σ is an unknown positive definite (p.d.) matrix, Θ is an unknown nonnegative definite (n.n.d.) matrix, and λ is a known positive scalar. For the estimation of Θ, a class of estimators of the form Θ̂(c,ϵ) = (c/λ){S1/q1 − ϵ(S2/q2)} (c ≥ 0, ϵ ≤ 1), uniformly better than the unbiased estimator Θ̂U = (1/λ){S1/q1 − S2/q2}, 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 Θ̂(c,ϵ) uniformly better than ΘU. It turns out that such an n.n.d. estimator exists only under restrictive conditions. However, for a suitable choice of c > 0, ϵ > 0, the estimator obtained by taking the positive part of Θ̂(c, ϵ) results in an n.n.d. estimator, say Θ̂(c, ϵ) +, that is uniformly better than Θ̂U. Numerical results indicate that in terms of mean squared error, Θ̂(c, ϵ) + performs much better than both Θ̂U and the restricted maximum likelihood estimator Θ̂REML of Θ. Similar results are also obtained for the nonnegative estimation of tr Θ and a′Θa, where a is an arbitrary nonzero vector. For estimating Σ, we have derived estimators that are claimed to be uniformly better than the unbiased estimator Σ̂U = 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 Σ.