Sharp simultaneous confidence intervals for the means of selected populations with application to microarray data analysis

Simultaneous inference for a large number, N, of parameters is a challenge. In some situations, such as microarray experiments, researchers are only interested in making inference for the K parameters corresponding to the K most extreme estimates. Hence it seems important to construct simultaneous confidence intervals for these K parameters. The naïve simultaneous confidence intervals for the K means (applied directly without taking into account the selection) have low coverage probabilities. We take an empirical Bayes approach (or an approach based on the random effect model) to construct simultaneous confidence intervals with good coverage probabilities. For N = 10,000 and K = 100, typical for microarray data, our confidence intervals could be 77% shorter than the naïve K-dimensional simultaneous intervals.