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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

Author

Listed:
  • Li Zhang

    (Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
    These authors contributed equally to this work.)

  • Ying-Ying Zhang

    (Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
    Chongqing Key Laboratory of Analytic Mathematics and Applications, Chongqing University, Chongqing 401331, China
    Department of Statistics, School of Mathematics and Statistics, Yunnan University, Kunming 650500, China
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Li Zhang & Ying-Ying Zhang, 2022. "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," Mathematics, MDPI, vol. 10(21), pages 1-27, October.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4005-:d:956394
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    References listed on IDEAS

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