Empirical Bayes false coverage rate controlling confidence intervals

. Benjamini and Yekutieli suggested that it is important to account for multiplicity correction for confidence intervals when only some of the selected intervals are reported. They introduced the concept of the false coverage rate (FCR) for confidence intervals which is parallel to the concept of the false discovery rate in the multiple-hypothesis testing problem and they developed confidence intervals for selected parameters which control the FCR. Their approach requires the FCR to be controlled in the frequentist's sense, i.e. controlled for all the possible unknown parameters. In modern applications, the number of parameters could be large, as large as tens of thousands or even more, as in microarray experiments. We propose a less conservative criterion, the Bayes FCR, and study confidence intervals controlling it for a class of distributions. The Bayes FCR refers to the average FCR with respect to a distribution of parameters. Under such a criterion, we propose some confidence intervals, which, by some analytic and numerical calculations, are demonstrated to have the Bayes FCR controlled at level q for a class of prior distributions, including mixtures of normal distributions and zero, where the mixing probability is unknown. The confidence intervals are shrinkage-type procedures which are more efficient for the ?is that have a sparsity structure, which is a common feature of microarray data. More importantly, the centre of the proposed shrinkage intervals reduces much of the bias due to selection. Consequently, the proposed empirical Bayes intervals are always shorter in average length than the intervals of Benjamini and Yekutieli and can be only 50% or 60% as long in some cases. We apply these procedures to the data of Choe and colleagues and obtain similar results.