Decision Approach and Empirical Bayes FCR-Controlling Interval for Mixed Prior Model

In this paper, I apply the decision theory and empirical Bayesian approach to construct confidence intervals for selected populations when true parameters follow a mixture prior distribution. A loss function with two tuning parameters $k_1$ and $k_2$ is coined to address the mixture prior. One specific choice of $k_2$ can lead to the procedure in Qiu and Hwang (2007); the other choice of $k_2$ provides an interval construction which controls the Bayes FCR. Both the analytical and extensive numerical simulation studies demonstrate that the new empirical Bayesian FCR controlling approach enjoys great length reduction. At the end, I apply different methods to a microarray data set. It turns out that the average length of the new approach is only 57% of that of Qiu and Hwang's procedure which controls the simultaneous non-coverage probability and 66% of that of Benjamini and Yekutieli (2005)'s procedure which controls the frequentist's FCR.