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

Abstract

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson 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 parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: 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 results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.

Keywords:

• Empirical Bayes estimators
• moment method
• maximum likelihood estimation (MLE) method
• Poisson distribution with a conjugate gamma prior
• Stein's loss function

2010:

• 62C12
• 62F10
• 62F15

Acknowledgments

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

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ying-Ying Zhang  http://orcid.org/0000-0002-6279-3662

Additional information

Funding

The research was supported by the Fundamental Research Funds for the Central Universities (2019CDXYST0016; 2018CDXYST0024), China Scholarship Council (201606055028), National Natural Science Foundation of China (11671060), and MOE project of Humanities and Social Sciences on the west and the border area (14XJC910001).