The Empirical Bayes Estimators of the Variance Parameter of the Normal Distribution with a Normal-Inverse-Gamma Prior under Stein's Loss Function

For the hierarchical normal and normal-inverse-gamma model, we derive the Bayesian estimator of the variance parameter in the normal distribution under Stein's loss function---a penalty function that treats gross overestimation and underestimation equally---and compute the associated Posterior Expected Stein's Loss (PESL). Additionally, we determine the Bayesian estimator of the same variance parameter under the squared error loss function, along with its corresponding PESL. We further develop empirical Bayes estimators for the variance parameter using a conjugate normal-inverse-gamma prior, employing both the method of moments and Maximum Likelihood Estimation (MLE). Through numerical simulations, we examine five key aspects: (1) the consistency of moment-based and MLE-based hyperparameter estimators; (2) the influence of κ₀ on quantities of interest as functions of the most recent observation; (3) two inequalities involving the Bayesian estimators and their respective PESL values; (4) the model's goodness-of-fit to simulated data; and (5) graphical representations of marginal densities under different hyperparameter settings. The simulation results demonstrate that MLEs outperform moment estimators in estimating hyperparameters, particularly with respect to consistency and model fit. Finally, we apply our methodology to real-world data on poverty levels---specifically, the percentage of individuals living below the poverty line---to validate and illustrate our theoretical findings.