Selecting The Exponential Population Having The Larger Guarantee Time With Unequal Sample Sizes

Abstract

Let π1 and π2 be two independent exponential populations, where the population πi has an unknown location parameter (or guarantee time) μi > 0 and known scale parameter σi > 0, i = 1, 2. Let μ[1] ⩽ μ[2] denote the ordered values of μ1 and μ2, and assume that the correct ordering between μ1 and μ2 is not known a priori. Consider the goal of selecting the population associated with μ[2] under the decision theoretic framework. We deal with the problem of finding the minimax selection rule under the 0-1 loss function (which takes the value 0 if correct selection is made and takes the value 1 if correct selection is not made) when (μ1, μ2) is known to lie in a certain subset of the parameter space, called the preference-zone. Based on independent random samples of (possibly) unequal sizes from the two populations, we propose a class of natural selection rules and find the minimax selection rule within this class. We call the minimax selection rule within this class to be the restricted minimax selection rule. This restricted minimax selection rule is shown to be globally minimax and generalized Bayes. A numerical study on the performance of various selection rules indicates that the minimax selection rule outperforms various natural selection rules.