Estimation of the Scale Parameter of the Selected Gamma Population Under the Entropy Loss Function

Abstract

Let X 1, X 2,…, X k be k (≥2) independent random variables from gamma populations Π1, Π2,…, Π k with common known shape parameter α and unknown scale parameter θ i , i = 1,2,…,k, respectively. Let X (i) denotes the ith order statistics of X 1,X 2,…,X k . Suppose the population corresponding to largest X (k) (or the smallest X (1)) observation is selected. We consider the problem of estimating the scale parameter θ M (or θ J ) of the selected population under the entropy loss function. For k ≥ 2, we obtain the Unique Minimum Risk Unbiased (UMRU) estimator of θ M (and θ J ). For k = 2, we derive the class of all linear admissible estimators of the form cX (2) (and cX (1)) and show that the UMRU estimator of θ M is inadmissible. The results are extended to some subclass of exponential family.