Estimation after subset selection from exponential populations

Let π1 πk be k independent exponential populations with means θ1 θkrespectively. Let Y1, ,Yk denote the sample means based on n independent observations from each population. For selecting a nonempty subset containing the best population (the one associated with the largest θi ) we consider the rule of Gupta which selects πi. if and only if where 0 < c < 1 is determined such that the probability of including the best population in the selected subset is at least P*, a pre-designed level. We consider the problem of estimating M, the mean of all θi 's in the selected subset, for k = 2. It is shown that the ‘natural’ estimator T is positively biased. An adjustment for bias of T is considered so that this adjusted estimator Tλ1 is unbiased when θ1 = θ2 The performances of T and Tλ1 are compared in terms of bias and mean square error for selected values of P*, n, and ρ - θ1./(θ1+θ2.