Estimation of the scale parameter of a selected gamma population with unequal shape parameters under stein loss function

The problem of estimation of the parameter of a selected population arises when we encounter with several populations and would like to estimate the parameter of the best (worst) selected population. Suppose X-i1, X-i2, ..., X-ini, i = 1, ..., k be k(>= 2) independent random samples drawn from populations Pi(1), Pi(2), ..., Pi(k), respectively, where observations from Pi(i) have a Gamma (alpha(i), theta(i))-distribution with unequal known shape parameters alpha(i), i = 1, ..., k. In this paper, we use a selection rule to select the best (worst) population, and estimate the best (worst) scale parameter theta(S) (theta(J)) of the selected population under the Stein loss function. The uniformly minimum risk unbiased (UMRU) estimators of theta(S) and theta(J) are obtained. A sufficient condition for inadmissibility of scale-invariant estimators of theta(S) (theta(J)) is obtained and it is shown that the UMRU estimator of theta(S) (theta(J)) is inadmissible. For k = 2, a sufficient condition for minimaxity of a given estimator of theta(S) (theta(J)) is obtained, and the generalized Bayes estimator of theta(S) is shown to be minimax. Finally, the risk functions of the various competing estimators are compared numerically, and a real data is provided to compute the proposed estimates and their expected risks.