The empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the MLE method

Abstract

Most of the samples in the real world are from the normal distributions with unknown mean and variance, for which it is common to assume a conjugate normal-inverse-gamma prior. We calculate the empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the Maximum Likelihood Estimation (MLE) method in two theorems. After that, we illustrate the two theorems for the monthly simple returns of the Shanghai Stock Exchange Composite Index.

Keywords:

• Empirical bayes estimators
• moment method
• maximum likelihood estimation (MLE) method
• normal distribution with normal-inverse-gamma prior
• non standardized student-t distribution

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 62C12
• 62F10
• 62F15

Acknowledgements

The authors are grateful to the editor, the associate editor, and the anonymous referees for their valuable comments and suggestions to improve this article.

Additional information

Funding

The research was supported by the Fundamental Research Funds for the Central Universities (CQDXWL-2012-004; 106112016CDJXY100002), 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).