Selecting the best of two gamma populations having unequal shape parameters

Consider two independent gamma populations π1π1 and π2π2, where the population πiπi has an unknown scale parameter θi>0θi>0 and known shape parameter αi>0,i=1,2αi>0,i=1,2. Assume that the correct ordering between θ1θ1 and θ2θ2 is not known a priori and let θ[1]≤θ[2]θ[1]≤θ[2] denote the ordered values of θ1θ1 and θ2θ2. Consider the goal of identifying (or selecting) the population associated with θ[2]θ[2], under the indifference-zone approach of Bechhofer (1954), when the quality of a selection rule is assessed in terms of the infimum of the probability of correct selection over the preference-zone. Under the decision-theoretic framework this goal is equivalent to that of finding the minimax selection rule when (θ1,θ2)(θ1,θ2) lies in the preference-zone and 0–1 loss function is used (which takes the value 0 if correct selection is made and takes the value 1 if correct selection is not made). Based on independent observations from the two populations, the minimax selection rule is derived. This minimax selection rule is shown to be generalized Bayes and admissible. Finally, using a numerical study, it is shown that the minimax selection rule outperforms various natural selection rules.