Empirical Bayes confidence intervals shrinking both means and variances

Summary.  We construct empirical Bayes intervals for a large number p of means. The existing intervals in the literature assume that variances     are either equal or unequal but known. When the variances are unequal and unknown, the suggestion is typically to replace them by unbiased estimators     . However, when p is large, there would be advantage in 'borrowing strength' from each other. We derive double-shrinkage intervals for means on the basis of our empirical Bayes estimators that shrink both the means and the variances. Analytical and simulation studies and application to a real data set show that, compared with the t -intervals, our intervals have higher coverage probabilities while yielding shorter lengths on average. The double-shrinkage intervals are on average shorter than the intervals from shrinking the means alone and are always no longer than the intervals from shrinking the variances alone. Also, the intervals are explicitly defined and can be computed immediately.