The empirical Bayes estimators of the parameter of the uniform distribution with an inverse gamma prior under Stein’s loss function

Abstract

For the hierarchical uniform and inverse gamma model, we calculate the Bayes posterior estimator of the parameter of the uniform 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 function and the corresponding PESL. Moreover, we obtain empirical Bayes estimators of the parameter of the uniform distribution by two methods. Note that the estimators of the hyperparameters of the model by the Maximum Likelihood Estimation (MLE) method are summarized in a theorem, whose proof involves the upper incomplete gamma function and a special case of the Meijer G-function. In numerical simulations, we address from four perspectives. First, we exemplify the two inequalities of the Bayes posterior estimators and the PESLs. Second, we illustrate that the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters. Third, we calculate the goodness-of-fit of the model for the simulated data. Fourth, we plot the marginal densities of the model for various hyperparameters. Finally, we utilize the current prices of the 300 component stocks of Shenzhen 300 Index to illustrate our theoretical studies.

Keywords:

• Empirical Bayes estimators
• Hierarchical uniform and inverse gamma model
• MLE method
• Moment method
• Stein’s loss function

2010 MSC:

• 62C12
• 62F10
• 62F15

Supporting information

Additional information for this article is available.

supplement_CSSC.pdf: this supplementary material contains some proofs of the article.

codes_CSSC.pdf: this supplementary material contains R codes, Matlab codes, and other codes of the article.

codes folder: this folder contains R codes, Matlab codes, and other codes used in the article.

Additional information

Funding

The research was supported by the Ministry of Education (MOE) Project of Humanities and Social Sciences on the West and the Border Area (20XJC910001), the National Social Science Fund of China (21XTJ001), and the National Natural Science Foundation of China (12001068; 72071019).