On F-Modelling based Empirical Bayes Estimation of Variances

We consider the problem of empirical Bayes estimation of multiple variances $\sigma_i^2$'s when providing with sample variances $S_i$'s. Assuming an arbitrary prior and an invariant loss function, the resultant Bayes estimator relies on $F(s^2)$, the marginal cumulative distribution function of the sample variances only. When replacing it by the empirical distribution function $F_N(s^2)$, we obtain an F-modelling based Empirical Bayes estimator of Variances (F-EBV), which converges to the corresponding Bayes version uniformly over a large set. We then construct confidence intervals for mean parameters using different variance estimators. It is shown that the intervals based on the F-EBV is the shortest among all the intervals which guarantee a desired coverage probability. Through real data analysis, we have shown that intervals based on the F-EBV lead to the smallest number of discordances, a desirable property when dealing with the current "replication crisis".