Abstract
Consider the generalized Poisson and the negative binomial model with mean parameter equal to kb, where \(k \ge 0\) is a count parameter and \(0< b < 1\) is a hyperparameter. We show that conditioning on counts from both models and assuming a uniform prior for k lead to the following Bayesian posterior distributions: (i) geometric for conditioning value of 0; (ii) extended negative binomial for conditioning value of 1; (iii) approximately extended Hurwitz–Lerch zeta distribution for conditioning value of 2 or more. Kullback–Leibler divergence for measuring the quality of the approximating distributions for some combinations of b and the mean–variance ratio is given.
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R package used: VGAM [23]. R codes for reproducing the analyses are available at https://github.com/Divo-Lee/R-Code/blob/main/Post_dist.R
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We thank two anonymous reviewers for their constructive comments which helped improve the clarity of the present work.
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TFK was involved in the conceptualisation; all authors contributed to the methodology, formal analysis, writing—original draft preparation and writing—review and editing.
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Communicated by Rosihan M. Ali.
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Li, H., Khang, T.F. Some Approximation Results for Bayesian Posteriors that Involve the Hurwitz–Lerch Zeta Distribution. Bull. Malays. Math. Sci. Soc. 46, 72 (2023). https://doi.org/10.1007/s40840-023-01463-9
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DOI: https://doi.org/10.1007/s40840-023-01463-9
Keywords
- Approximation
- Generalized Poisson distribution
- Hurwitz–Lerch zeta distribution
- Negative binomial distribution
- Posterior distribution