  Consider two independent exponential populations having different unknown
location parameters and a common unknown scale parameter. Call the population
associated with the larger location parameter as the "best" population and the
population associated with the smaller location parameter as the "worst"
population. For the goal of selecting the best (worst) population a natural
selection rule, that has many optimum properties, is the one which selects the
population corresponding to the larger (smaller) minimal sufficient statistic.
In this article, we consider the problem of estimating the location parameter
of the population selected using this natural selection rule. For estimating
the location parameter of the selected best population, 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. We provide similar results for the
problem of estimating the location parameter of the selected population when
the selection goal is that of selecting the worst exponential population.
Finally, we provide a simulation study to compare, numerically, the
performances of some of the proposed estimators.
