The Bayesian Posterior and Marginal Densities of the Hierarchical Gamma–Gamma, Gamma–Inverse Gamma, Inverse Gamma–Gamma, and Inverse Gamma–Inverse Gamma Models with Conjugate Priors

Li Zhang and Ying-Ying Zhang ()
Additional contact information
Li Zhang: Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
Ying-Ying Zhang: Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Mathematics, 2022, vol. 10, issue 21, 1-27

Abstract: Positive, continuous, and right-skewed data are fit by a mixture of gamma and inverse gamma distributions. For 16 hierarchical models of gamma and inverse gamma distributions, there are only 8 of them that have conjugate priors. We first discuss some common typical problems for the eight hierarchical models that do not have conjugate priors. Then, we calculate the Bayesian posterior densities and marginal densities of the eight hierarchical models that have conjugate priors. After that, we discuss the relations among the eight analytical marginal densities. Furthermore, we find some relations among the random variables of the marginal densities and the beta densities. Moreover, we discuss random variable generations for the gamma and inverse gamma distributions by using the R software. In addition, some numerical simulations are performed to illustrate four aspects: the plots of marginal densities, the generations of random variables from the marginal density, the transformations of the moment estimators of the hyperparameters of a hierarchical model, and the conclusions about the properties of the eight marginal densities that do not have a closed form. Finally, we illustrate our method by a real data example, in which the original and transformed data are fit by the marginal density with different hyperparameters.

Keywords: conjugate prior; gamma and inverse gamma distribution; hierarchical model and mixture distribution; marginal density; posterior density (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/21/4005/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/21/4005/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:21:p:4005-:d:956394

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().