ADMISSIBLITY OF SIMULTANEOUS ESTIMATION OF TWO VARIANCE COMPONENTS

Consider a variance-oomponent model \left\\beginarrayl E \undersetp \times 1Y=X \undersetp \times pX \cdot \undersetp \times 1\beta \\ D Y=\sigma_1^2 V_1+\sigma_2^2 V_2, \endarray\right. where \beta \in R^p, \sigma_1^2>0, and \sigma_2^2>0 are all unknown, X, V_1>0 and V_2>0 are all knwon, r(X)=p.The author estimates simultaneously \left(\sigma_1^2, \sigma_2^2\right. ) and oonsiders the estimator class \mathscrX=\left\d\left(A_1, A_2\right)=\left(Y^\prime A_1 Y, Y^\prime A_2 Y\right), A_1 \geqslant 0, A_2 \geqslant 0\right\ . The loss function is L\left(d\left(A_1, A_2\right),\left(\sigma_1^2, \sigma_2^2\right)\right)=\frac1\sigma_1^4\left(Y^\prime A_1 Y-\sigma_1^2\right)^2+\frac1\sigma_2^4\left(Y^\prime A_2 Y-\sigma_2^2\right)^2 . This paper gives both a sufficient condition and a necessary condition for d\left(A_1, A_2\right) to be a \mathscrD-admissible estimator under the restriction V_1=V_2, and a sufficient oondition for d\left(A_1, \Lambda_2\right) to be a 0 -admissible estimator without this restriotion.