IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i21p4005-d956394.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/21/4005/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/10/21/4005/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Stefano Cabras, 2021. "A Bayesian-Deep Learning Model for Estimating COVID-19 Evolution in Spain," Mathematics, MDPI, vol. 9(22), pages 1-18, November.
    2. Ji-Ping Wang, 2010. "Estimating species richness by a Poisson-compound gamma model," Biometrika, Biometrika Trust, vol. 97(3), pages 727-740.
    3. Ying-Ying Zhang & Yu-Han Xie & Wen-He Song & Ming-Qin Zhou, 2020. "The Bayes rule of the parameter in (0,1) under Zhang’s loss function with an application to the beta-binomial model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(8), pages 1904-1920, April.
    4. Shujin Wu, 2022. "Poisson-Gamma mixture processes and applications to premium calculation," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 51(17), pages 5913-5936, September.
    5. Dümbgen, Lutz & Wellner, Jon A., 2020. "The density ratio of Poisson binomial versus Poisson distributions," Statistics & Probability Letters, Elsevier, vol. 165(C).
    6. Jakimauskas, Gintautas & Sakalauskas, Leonidas, 2016. "Note on the singularity of the Poisson–gamma model," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 86-92.
    7. Ying-Ying Zhang & Yu-Han Xie & Wen-He Song & Ming-Qin Zhou, 2018. "Three strings of inequalities among six Bayes estimators," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(8), pages 1953-1961, April.
    8. Ying-Ying Zhang, 2017. "The Bayes rule of the variance parameter of the hierarchical normal and inverse gamma model under Stein's loss," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(14), pages 7125-7133, July.
    9. Ehm, Werner, 1991. "Binomial approximation to the Poisson binomial distribution," Statistics & Probability Letters, Elsevier, vol. 11(1), pages 7-16, January.
    10. repec:dau:papers:123456789/1908 is not listed on IDEAS
    11. Ying-Ying Zhang & Teng-Zhong Rong & Man-Man Li, 2019. "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," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(9), pages 2286-2304, May.
    12. Ying-Ying Zhang & Naitee Ting, 2021. "Can the Concept Be Proven?," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 13(1), pages 160-177, April.
    13. R. Vijayaraghavan & K. Rajagopal & A. Loganathan, 2008. "A procedure for selection of a gamma-Poisson single sampling plan by attributes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 35(2), pages 149-160.
    14. Kristin A. Duncan & Jonathan L. Wilson, 2008. "A Multinomial‐Dirichlet Model for Analysis of Competing Hypotheses," Risk Analysis, John Wiley & Sons, vol. 28(6), pages 1699-1709, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. repec:jss:jstsof:40:i09 is not listed on IDEAS
    2. Arun G. Chandrasekhar & Robert Townsend & Juan Pablo Xandri, 2018. "Financial Centrality and Liquidity Provision," NBER Working Papers 24406, National Bureau of Economic Research, Inc.
    3. David M. Phillippo & Sofia Dias & A. E. Ades & Mark Belger & Alan Brnabic & Alexander Schacht & Daniel Saure & Zbigniew Kadziola & Nicky J. Welton, 2020. "Multilevel network meta‐regression for population‐adjusted treatment comparisons," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 183(3), pages 1189-1210, June.
    4. Loganathan A. & Vijayaraghavan R. & Rajagopal K., 2010. "Designing Single Sampling Plans by Variables Using Predictive Distribution," Stochastics and Quality Control, De Gruyter, vol. 25(2), pages 301-316, January.
    5. Marie Ernst & Yvik Swan, 2022. "Distances Between Distributions Via Stein’s Method," Journal of Theoretical Probability, Springer, vol. 35(2), pages 949-987, June.
    6. Zhang, Hongmei & Ghosh, Kaushik & Ghosh, Pulak, 2012. "Sampling designs via a multivariate hypergeometric-Dirichlet process model for a multi-species assemblage with unknown heterogeneity," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2562-2573.
    7. Róbert Pethes & Levente Kovács, 2023. "An Exact and an Approximation Method to Compute the Degree Distribution of Inhomogeneous Random Graph Using Poisson Binomial Distribution," Mathematics, MDPI, vol. 11(6), pages 1-24, March.
    8. Seungchul Baek & Junyong Park, 2022. "A computationally efficient approach to estimating species richness and rarefaction curve," Computational Statistics, Springer, vol. 37(4), pages 1919-1941, September.
    9. Chun-Huo Chiu & Yi-Ting Wang & Bruno A. Walther & Anne Chao, 2014. "An improved nonparametric lower bound of species richness via a modified good–turing frequency formula," Biometrics, The International Biometric Society, vol. 70(3), pages 671-682, September.
    10. Vijayaraghavan R. & Sakthivel K. M., 2011. "Chain Sampling Inspection Plans Based on Bayesian Methodology," Stochastics and Quality Control, De Gruyter, vol. 26(2), pages 173-187, January.
    11. Aihua Xia & Fuxi Zhang, 2009. "Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance," Journal of Theoretical Probability, Springer, vol. 22(2), pages 294-310, June.
    12. Vijayaraghavan R. & Sakthivel K. M., 2010. "Selection of Bayesian Double Sampling Inspection Plans by Attributes with Small Acceptance Numbers," Stochastics and Quality Control, De Gruyter, vol. 25(2), pages 207-220, January.
    13. Zhang, Yazhe, 2016. "Binomial approximation for sum of indicators with dependent neighborhoods," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 146-154.
    14. Arun Chandrasekhar & Robert Townsend & Juan Pablo Pablo Xandri, 2019. "Financial Centrality and the Value of Key Players," Working Papers 2019-26, Princeton University. Economics Department..
    15. Vydas Čekanavičius & Bero Roos, 2006. "Compound Binomial Approximations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 187-210, March.
    16. Pérez-González, Carlos J. & Fernández, Arturo J. & Kohansal, Akram, 2020. "Efficient truncated repetitive lot inspection using Poisson defect counts and prior information," European Journal of Operational Research, Elsevier, vol. 287(3), pages 964-974.
    17. López, Fernando & Matilla-García, Mariano & Mur, Jesús & Marín, Manuel Ruiz, 2010. "A non-parametric spatial independence test using symbolic entropy," Regional Science and Urban Economics, Elsevier, vol. 40(2-3), pages 106-115, May.
    18. Chee, Chew-Seng & Wang, Yong, 2016. "Nonparametric estimation of species richness using discrete k-monotone distributions," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 107-118.
    19. Christophe Ley & Gesine Reinert & Yvik Swan, 2014. "Approximate Computation of Expectations: the Canonical Stein Operator," Working Papers ECARES ECARES 2014-36, ULB -- Universite Libre de Bruxelles.
    20. Greene, Evan & Wellner, Jon A., 2016. "Finite sampling inequalities: An application to two-sample Kolmogorov–Smirnov statistics," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3701-3715.
    21. Biscarri, William & Zhao, Sihai Dave & Brunner, Robert J., 2018. "A simple and fast method for computing the Poisson binomial distribution function," Computational Statistics & Data Analysis, Elsevier, vol. 122(C), pages 92-100.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4005-:d:956394. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.