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Given p independent normal populations, we consider the problem of estimatingthe mean of those populations, that based on the observed data, give thestrongest signals. We explicitly condition on the ranking of the sample means,and consider a constrained conditional maximum likelihood (CCMLE) approach,avoiding the use of any priors and of any sparsity requirement between thepopulation means. Our results show that if the observed means are too closetogether, we should in fact use the grand mean to estimate the mean of thepopulation with the larger sample mean. If they are separated by more than acertain threshold, we should shrink the observed means towards each other. Asintuition suggests, it is only if the observed means are far apart that weshould conclude that the magnitude of separation and consequent ranking are notdue to chance. Unlike other methods, our approach does not need to pre-specifythe number of selected populations and the proposed CCMLE is able to performsimultaneous inference. Our method, which is conceptually straightforward, canbe easily adapted to incorporate other selection criteria.  Selected populations, Maximum likelihood, Constrained MLE, Post-selectioninference

A Constrained Conditional Likelihood Approach for Estimating the Means of Selected Populations