Equivariant estimation of the selected location parameter

Let $$X_1$$ X 1 and $$X_2$$ X 2 denote the smallest order statistics based on random samples of the same size (n) from two exponential populations having different unknown location parameters and a common unknown scale parameter. Call the population associated with the larger (smaller) location parameter as the “better" (“worse") population. For selecting the better (worse) population, consider the natural selection rule, that is known to possess several optimum properties, which selects the population corresponding to $$\max \left\{ X_1,X_2\right\} $$ max X 1 , X 2 ( $$\min \left\{ X_1,X_2\right\} $$ min X 1 , X 2 ) as the better (worse) population. After the selection of a population using the aforementioned natural selection rule, a problem of practical interest is to estimate the location parameter of the selected population, which is a random parameter. In this article we take up this estimation problem. We derive the uniformly minimum variance unbiased estimator (UMVUE) and show that the analogue of the best affine equivariant estimators (BAEEs) of location parameters is a generalized Bayes estimator. We provide some admissibility and minimaxity results for estimators in the class of linear, affine and permutation equivariant estimators, under the criterion of scaled mean squared error. We also derive a sufficient condition for inadmissibility of an arbitrary affine and permutation equivariant estimator. Finally, we provide a simulation study to compare numerically, the performance of some of the proposed estimators.