1. The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function
- Author
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Ze-Yu Wang, Wen Mi, Ying-Ying Zhang, and Zheng-Min Duan
- Subjects
Statistics and Probability ,021103 operations research ,Applied Mathematics ,0211 other engineering and technologies ,Estimator ,02 engineering and technology ,Function (mathematics) ,Poisson distribution ,01 natural sciences ,Statistics::Computation ,010104 statistics & probability ,symbols.namesake ,Bayes' theorem ,Modeling and Simulation ,symbols ,Statistics::Methodology ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Conjugate - 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.
- Published
- 2019