The empirical Bayes estimators of the rate parameter of the inverse gamma distribution with a conjugate inverse gamma prior under Stein's loss function

Abstract

For the hierarchical inverse gamma and inverse gamma model, we calculate the Bayes posterior estimator of the rate parameter of the inverse gamma 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 rate parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the rate parameter of the inverse gamma distribution with a conjugate inverse gamma prior by the moment and MLE methods. In numerical simulations, we have illustrated four aspects: Consistency, goodness-of-fit, comparison, and marginal densities. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, the model could potentially be used to fit right skewed data, not left-skewed data.

Keywords:

• Empirical Bayes estimators
• moment method
• maximum likelihood estimation (MLE) method
• hierarchical inverse gamma and inverse gamma model
• Stein's loss function

2010 Mathematics Subject Classifications:

• 62C12
• 62F10
• 62F15

The authors would like to thank the Editor, the Associate Editor, and the referees for their constructive comments, which have led to a substantial improvement of the article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research was supported by the Fundamental Research Funds for the Central Universities (2019CDXYST0016; 2018CDXYST0024), MOE project of Humanities and Social Sciences on the west and the border area (20XJC910001), China Scholarship Council (201606055028), National Natural Science Foundation of China (11671060), and Chongqing Key Laboratory of Analytic Mathematics and Applications.