On some inadmissibility results for the scale parameters of selected gamma populations

Let X1 and X2 be two independent gamma random variables, having unknown scale parameters λ1 and λ2, respectively, and common known shape parameter p(>0). Define, M=1, if X1X2 and J=3-M. We consider the componentwise estimation of the random parameters λM and λJ, under the squared error loss functions L1(λ̲,δ1)=(δ1-λM)2 and L2(λ̲,δ2)=(δ2-λJ)2, respectively. We derive a general result which provides sufficient conditions for a scale and permutation invariant estimator of λM (or λJ) 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, Xi/(p+1),i=1,2, is the best scale invariant estimator of λi for the component problem, estimators δ1,c1(X̲)=XM/(p+1) and δ2,c1(X̲)=XJ/(p+1) are the natural analogs of X1/(p+1) and X2/(p+1) for estimating λM and λJ, respectively. From the general result we derive, it follows that the natural estimators δ1,c1(X̲)=XM/(p+1) and δ2,c1(X̲)=XJ/(p+1) are inadmissible for estimating λM and λJ, respectively, within the class of scale and permutation invariant estimators and the dominating scale and permutation invariant estimators are obtained. For the estimation of λJ, 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 δ2,c1(·), are obtained. It is also established that any estimator which is a constant multiple of XJ is inadmissible for estimating λJ. For 0