Sankhy¯a: The Indian Journal of Statistics 1994, Volume 56, Series B, Pt. 3, pp. 344-355 ESTIMATION OF THE AVERAGE WORTH OF THE SELECTED SUBSET OF GAMMA POPULATIONS By NEERAJ MISRA Panjab University SUMMARY. Let X 1 ,...,X k be k (≥ 2) independent random variables representing the pop- ulations Π 1 ,..., Π k , respectively and suppose that the random variable X i has a gamma distri- bution with known shape parameter p and unknown scale parameter θ i ,i =1,...,k. For the goal of selecting a nonempty subset of {Π 1 ,..., Π k }, containing the best population (one associated with max (θ 1 ,...,θ k )), we consider the decision rule which selects Π i if and only if X i ≥ c max (X 1 ,...,X k ), i = 1, ...k, where 0 <c ≤ 1 is chosen so that the probability of including the best population in the selected subset is at least P ∗ (1/k ≤ P ∗ < 1), a pre-assigned level. We consider the problem of estimating W , the mean of all θ i ’s in the selected subset for a general k (≥ 2). The uniformly minimum variance unbiased estimator (which is also the uniformly minimum risk unbi- ased estimator) is obtained and it is established that the natural estimator of W is inadmissible for the squared error loss function. Better estimators are obtained. 1. Introduction Let Π 1 ,..., Π k be k (≥ 2) independent populations such that the random vari- able X i representing the population Π i has a probability density function (p.d.f) : f (x; θ i )= 1 θ p i Γ(p) x p−1 exp (− x θ i ), x> 0, θ i > 0, p> 0, i =1,...,k. We assume that θ 1 , ....,θ k are unknown and p is known. Let X = {x : x = (x 1 , ..., x k ), x i > 0,i =1, ..., k} and Ω = {θ : θ =(θ 1 , ....,θ k ),θ i > 0, i =1, ..., k} denote the sample space and the parameter space, respectively. Assume that the correct pairing between the ordered and the unordered θ i ’s is not known, and con- sider the goal of selecting a nonempty subset S of {1, ..., k} such that S contains the index of the “best” population (one associated with max (θ 1 , .....,θ k )) with probability at least equal to P ∗ (1/k ≤ P ∗ < 1), a pre- Paper received. October 1992; revised August 1993. AMS subject classification. (1980) Primary 62F07, secondary 62F10. Key words and phrases. Subset selection, worth of the selected subset, correct selection, natu- ral estimator, inadmissible estimator, uniformly minimum variance unbiased estimator, uniformly minimum risk unbiased estimator.