Resumen
Let X1 and X2 be two independent gamma random variables, having unknown scale parameters ?1 and ?2, 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 ?M and ?J, under the squared error loss functions L1(?_,d1)=(d1-?M)2 and L2(?_,d2)=(d2-?J)2, respectively. We derive a general result which provides sufficient conditions for a scale and permutation invariant estimator of ?M (or ?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, Xi/(p+1),i=1,2, is the best scale invariant estimator of ?i for the component problem, estimators d1,c1(X_)=XM/(p+1) and d2,c1(X_)=XJ/(p+1) are the natural analogs of X1/(p+1) and X2/(p+1) for estimating ?M and ?J, respectively. From the general result we derive, it follows that the natural estimators d1,c1(X_)=XM/(p+1) and d2,c1(X_)=XJ/(p+1) are inadmissible for estimating ?M and ?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 ?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 d2,c1(·), are obtained. It is also established that any estimator which is a constant multiple of XJ is inadmissible for estimating ?J. For 0