Abstract:
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Researchers are often interested in making inference on one or a few "best" treatments out of p given treatments. This problem is referred to as post-selection inference. Substantial research has been done on constructing point estimates and confidence sets for a multivariate normal mean vector, but there are very few works on selection. Often, out of all available estimates, the ones that are minimax and/or admissible are preferred. It was proven by Sacrowitz and Samuel-Cahn (1980) that X_1, the first order statistic, is minimax for estimating the selected mean for p< =3, but it is not minimax for p>3, but the question whether it is admissible is still open. Following the arguments of Berger (1976) and Maruyama (2009) we prove that X_1 is admissible for p< 4 and some bias correction is needed for p>=4. The bias corrected estimate, the generalized Bayes estimate under some prior, will be admissible for p>=4. We also provide a comparison of this admissible estimator and other estimators proposed in the literature, including the estimator proposed in Reid et al. (2014) and a generalized Bayes estimator under the horseshoe prior.
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