For the probability parameter of the beta-negative binomial model, we recommend and analytically calculate the Bayes estimator under Zhang's loss function which penalizes gross overestimation and gross underestimation equally. This estimator minimizes the posterior expected Zhang's loss (PEZL). We also calculate the usual Bayes estimator under the squared error loss function. Moreover, we obtain the PEZLs evaluated at the two Bayes estimators. After that, we show two theorems about the estimators of the hyperparameters of the beta-negative binomial model by the moment method and the maximum likelihood estimation (MLE) method, respectively. Hence, the empirical Bayes estimator of the probability parameter under Zhang's loss function is obtained with the hyperparameters estimated from the two theorems. In the numerical simulations, we have illustrated three things. Firstly, we have exemplified the two inequalities of the Bayes estimators and the PEZLs. Secondly, we have illustrated that the moment estimators and the maximum likelihood estimators (MLEs) are consistent estimators of the hyperparameters. Thirdly, we have calculated the goodness-of-fit of the beta-negative binomial model to the simulated data.
Finally, we consider four cases to fit a real insurance claims data by utilizing the beta-negative binomial model.
The research was supported by the Fundamental Research Funds for the Central Universities (Grant Nos. 2020CDJSK02ZH03; 2021CDSKXYJG012), the MOE project of Humanities and Social Sciences on the west and the border area (Grant Nos. 20XJC910001; 14XJC910001), the China Scholarship Council (Grant No. 201606055028) and the National Natural Science Foundation of China (Grant No. 11671060).
ZHOU Mingqin,ZHANG Yingying,SUN Ya等. The Empirical Bayes Estimators of the Probability Parameter of the Beta-Negative Binomial Model under Zhang's Loss Function[J]. CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST, 2021, 37(5): 478-494.