The empirical Bayes estimators of the variance parameter of the normal distribution with a conjugate inverse gamma prior under Stein’s loss function

Abstract

For the hierarchical normal and inverse gamma model, we calculate the Bayes posterior estimator of the variance parameter of the normal distribution under Stein’s loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein’s Loss (PESL). We also obtain the Bayes posterior estimator of the variance parameter under the squared error loss function and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the variance parameter of the normal distribution with a conjugate inverse gamma prior by two methods. In numerical simulations, we have illustrated five aspects: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data; the numerical comparisons of the Bayes posterior estimators and the PESLs of the oracle, moment, and MLE methods; and the plots of the marginal densities for various hyperparameters. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we utilize the %bodyfat data of 250 men of various ages to illustrate our theoretical studies.

Keywords:

• Empirical Bayes estimators
• moment method
• Maximum Likelihood Estimation (MLE) method
• normal distribution with a conjugate inverse gamma prior
• non standardized student-t distribution

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 62C12
• 62F10
• 62F15

Additional information

Funding

The research was supported by the Ministry of Education (MOE) Project of Humanities and Social Sciences on the West and the Border Area (20XJC910001), the National Social Science Fund of China (21XTJ001), and the National Natural Science Foundation of China (12001068; 72071019).