Confidence intervals for the means of the selected populations

Consider an experiment in which p independent populations pi(i), with corresponding unknown means theta(i), are available, and suppose that for every 1 <= i <= p, we can obtain a sample X-i1, . . . , X-in from pi(i). In this context, researchers are sometimes interested in selecting the populations that yield the largest sample means as a result of the experiment, and then estimate the corresponding population means theta(i). In this paper, we present a frequentist approach to the problem and discuss how to construct simultaneous confidence intervals for the means of the k selected populations, assuming that the populations pi(i) are independent and normally distributed with a common variance sigma(2). The method, based on the minimization of the coverage probability, obtains confidence intervals that attain the nominal coverage probability for any p and k, taking into account the selection procedure.