TY - JOUR
T1 - Empirical bayes minimax estimators of matrix normal means for arbitrary quadratic loss and unknown covariance matrix
AU - Shieh, Gwowen
PY - 1993/4
Y1 - 1993/4
N2 - Let X = (X1,…, XK), where Xi are mutually independent p-variate (K > p+1) normal vectors with unknown means θiand unknown positive definite variance-covariance matrix V. Assume the statistic V is available for estimating V, where V has a Wishart distribution WP(n, V)/(n+p+1), n> p+1, and is independent of X. It is desired to estimate θ = (θ1,…,θK) under the quadratic loss LQ*(θ, θ) = tr{(θ - θ)1Q*(θ - θ)}, where Q* = V-1/2QV-1/2, V =V1/2V1/2, and Q is a known positive definite matrix chosen by the researcher. The LQ* loss includes the widely used loss L(θ, θ) = tr{(θ - θ)1V-1(θ - θ)} as a special case. It is shown that under some specifications of τ(V,S), a symmetric pxp matrix, the proposed empirical Bayes estimator (Ip - (VS-1 τ(V, S))X dominates the maximum likelihood estimator X and is minimax under the LQ* loss. Unlike the previous work on the estimation of vector normal means under quadratic losses with a weight matrix Q, the proposed empirical Bayes minimax estimators are structurally free of Q and the minimaxity holds for a class of quadratic loss functions LQ*. The simulated risks of several competing EB estimators are considered, and the risk improvement of these estimators over the sample mean is calculated.
AB - Let X = (X1,…, XK), where Xi are mutually independent p-variate (K > p+1) normal vectors with unknown means θiand unknown positive definite variance-covariance matrix V. Assume the statistic V is available for estimating V, where V has a Wishart distribution WP(n, V)/(n+p+1), n> p+1, and is independent of X. It is desired to estimate θ = (θ1,…,θK) under the quadratic loss LQ*(θ, θ) = tr{(θ - θ)1Q*(θ - θ)}, where Q* = V-1/2QV-1/2, V =V1/2V1/2, and Q is a known positive definite matrix chosen by the researcher. The LQ* loss includes the widely used loss L(θ, θ) = tr{(θ - θ)1V-1(θ - θ)} as a special case. It is shown that under some specifications of τ(V,S), a symmetric pxp matrix, the proposed empirical Bayes estimator (Ip - (VS-1 τ(V, S))X dominates the maximum likelihood estimator X and is minimax under the LQ* loss. Unlike the previous work on the estimation of vector normal means under quadratic losses with a weight matrix Q, the proposed empirical Bayes minimax estimators are structurally free of Q and the minimaxity holds for a class of quadratic loss functions LQ*. The simulated risks of several competing EB estimators are considered, and the risk improvement of these estimators over the sample mean is calculated.
KW - Wishart identity
KW - empirical Bayes
KW - frequentist risks
KW - matrix normal means
KW - minimax
KW - quadratic loss
UR - http://www.scopus.com/inward/record.url?scp=0348023747&partnerID=8YFLogxK
U2 - 10.1524/strm.1993.11.4.317
DO - 10.1524/strm.1993.11.4.317
M3 - Article
AN - SCOPUS:0348023747
SN - 2193-1402
VL - 11
SP - 317
EP - 342
JO - Statistics and Risk Modeling
JF - Statistics and Risk Modeling
IS - 4
ER -