Journal of Statistical Planning and Inference vol:136 issue:7 pages:2340-2351
Let X-1 and X-2 be two independent gamma random variables, having unknown scale parameters). lambda(1) and lambda(2), respectively. and common known shape parameter p (> 0). Define, M = 1, if X-1 < X-2, M =2, if X-1 > X-2 and J = 3 - M. We consider the componentwise estimation of the random parameters lambda(M) and lambda(J) under the squared error loss functions L-1 (lambda, delta(1)) = (delta(1) - lambda(M))(2) and L-2(lambda delta(2)) = (delta - lambda(J))(2), respectively. We derive a general result which provides sufficient conditions for a scale and permutation invariant estimator of lambda(M) (or lambda(J)) to be inadmissible under the squared error loss function. In situations where these sufficient conditions are satisfied, this result also provides dominating estimators. Since, under the squared error loss function, X-i/(p + 1), i = 1, 2, is the best scale invariant estimator of lambda(i) for the component problem, estimators delta 1.(C1) ((X) under bar) = X-M/(p + 1) and delta(2),c(1) (S) = X-J/(p + 1) are the natural analogs of X-1(p + 1) and X-2/(p + 1) for estimating lambda(M) and lambda(J) respectively. From the general result we derive, it follows that the natural estimators delta(1).(c1) ((X) under bar) = X-M(p + 1) and delta(2).c(1)((X) under bar) = X-J/(p + 1) are inadmissible for estimating lambda(M) and lambda(J), respectively, within the class of scale and permutation invariant estimators and the dominating scale and permutation invariant estimators are obtained. For the estimation of lambda(J), improvements over various estimators derived by Vellaisamy [1992. Inadmissibility results for the selected scale parameter. Ann. Statist. 20, 2183-2191], which are known to dominate the natural estimator delta(2.c1), ((.)) are obtained. It is also established that any estimator which is a constant multiple of X-J is inadmissible for estimating lambda(J). For 0 < p < 1, an open problem concerning the inadmissibility of the uniformly minimum variance unbiased estimator of lambda(J) is resolved. Finally, we derive another general result which provides relations between the problems of estimation after selection and estimation of ranked parameters. Applications of this result to the problem of estimating the largest scale parameter of k (>= 2) gamma populations are provided.