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 provided with sample variances $s_i^2$'s. Assuming an arbitrary prior on $\sigma_i^2$'s, we derive different versions of the Bayes estimators using different loss functions. For one particular loss function, the resultant Bayes estimator relies on $F(s^2)$, the marginal cumulative distribution function of the sample variances only. When replacing it with the empirical distribution function $F_N(s^2)$, we obtain an empirical Bayes version called {\bf F}-modeling based {\bf E}mpirical {\bf B}ayes estimator of {\bf V}ariances (F-EBV). It is shown theoretically that F-EBV converges to the corresponding Bayes version {\it uniformly} over a large set. It can be used for post-selection estimation and the {\it finite Bayes} inference problem. We have demonstrated the advantages of F-EBV through extensive simulations and real data analysis.