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

Abstract

Let X 1 and X 2 be two independent gamma random variables, having unknown scale parameters λ 1 and λ 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 λ M and λ J , under the squared error loss functions L 1 ( λ ̲ , δ 1 ) = ( δ 1 - λ M ) 2 and L 2 ( λ ̲ , δ 2 ) = ( δ 2 - λ J ) 2 , respectively. We derive a general result which provides sufficient conditions for a scale and permutation invariant estimator of λ M (or λ 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 λ i for the component problem, estimators δ 1 , c 1 ( X ̲ ) = X M / ( p + 1 ) and δ 2 , c 1 ( X ̲ ) = X J / ( p + 1 ) are the natural analogs of X 1 / ( p + 1 ) and X 2 / ( p + 1 ) for estimating λ M and λ J , respectively. From the general result we derive, it follows that the natural estimators δ 1 , c 1 ( X ̲ ) = X M / ( p + 1 ) and δ 2 , c 1 ( X ̲ ) = X J / ( p + 1 ) are inadmissible for estimating λ M and λ 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 λ 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 δ 2 , c 1 ( · ) , are obtained. It is also established that any estimator which is a constant multiple of X J is inadmissible for estimating λ J . For 0 < p < 1 , an open problem concerning the inadmissibility of the uniformly minimum variance unbiased estimator of λ 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.