张损失函数下贝塔负二项模型概率参数的经验贝叶斯估计量

The Empirical Bayes Estimators of the Probability Parameter of the Beta-Negative Binomial Model under Zhang's Loss Function

对于贝塔负二项模型的概率参数,我们推荐并解析地计算了张损失函数下的贝叶斯估计量张损失函数对总的高估和总的低估有相等的惩罚.该估计量使后验期望张损失最小化.在平方误差损失函数下,我们还计算了常用的贝叶斯估计量.此外,我们得到了后验期望张损失在两个贝叶斯估计量下的估计.然后,我们分别用矩法和极大似然估计法给出了贝塔负二项模型超参数估计量的两个定理.因此,利用这两个定理估计的超参数,得到了张损失函数下概率参数的经验贝叶斯估计量.在数值模拟中,我们例证了三件事.首先,我们举例说明了贝叶斯估计量和后验期望张损失的两个不等式.其次,我们证明了矩估计量和极大似然估计量是超参数的一致估计量.第三,我们计算了贝塔负二项模型与模拟数据的拟合优度.最后,我们利用贝塔负二项式模型,考虑四种情况来拟合一个真实的保险理赔数据.

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.