On admissibility of variance components estimates (Q1083814)

Suppose that there is a variance components model \[ E_{n\times 1}=_{n\times p}\beta_{p\times 1},\quad DY=\sigma^ 2_ 2V_ 1+\sigma^ 2_ 2V_ 2, \] where \(\beta\), \(\sigma^ 2_ 1\) and \(\sigma^ 2_ 2\) are all unknown, X, \(V_ 1>0\) and \(V_ 2>0\) are all known, \(r(X)<n\). The author estimates simultaneously \((\sigma^ 2_ 1,\sigma^ 2_ 2)\). Estimators are restricted to the class \({\mathcal D}=\{d(A_ 1,A_ 2)=(Y'A_ 1Y,Y'A_ 2Y),A_ 1\geq 0,A_ 2\geq 0\}\). Suppose that the loss function is \[ L(d(A_ 1,A_ 2),(\sigma^ 2_ 1,\sigma^ 2_ 2))=(1/\sigma^ 4_ 1)(Y'A_ 1Y-\sigma^ 2_ 1)+(1/\sigma^ 4_ 2)(Y'A_ 2Y-\sigma^ 2_ 2)^ 2. \] This paper gives a necessary and sufficient condition for \(d(A_ 1,A_ 2)\) to be an equivariant \({\mathcal D}\)-admissible estimator under the restriction \(V_ 1=V_ 2\), and a sufficient condition and a necessary condition for \(d(A_ 1,A_ 2)\) to be equivariant \({\mathcal D}\)-admissible without the restriction.