Equivariant estimation following selection from two normal populations having common unknown variance

Abstract

For estimating the mean of the selected normal population (the one corresponding to the larger sample mean), we prove some admissibility and minimaxity results for the estimators belonging to the class of linear, affine and permutation equivariant estimators under the criteria of scaled mean squared error and scaled absolute bias. We derive a sufficient condition for inadmissibility of any affine and permutation equivariant estimator under the scaled mean squared error criterion. Consequently, all linear, affine and permutation equivariant estimators (except the naive estimator, larger of the two sample means) and some estimators proposed in Hsieh [On estimating the mean of the selected population with unknown variance. Commun Stat-Theor Methods. 1981;10(18):1869–1878] become inadmissible under scaled mean squared error criterion. We further show that under the scaled mean squared error criterion, the naive estimator is minimax, admissible and generalized Bayes with respect to the Jeffreys' prior. We analyse a real data set for illustration purpose and provide a simulation study to compare the performances of various competing estimators.