Bayesian Estimation and Prediction for Zero-Inflated Discrete Weibull Distribution

Monthira Duangsaphon; Kamon  Budsaba; Sudarat  Nidsunkid; Dusit  Chaiprasithikul

Authors

Monthira Duangsaphon Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani, Thailand
Kamon Budsaba Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani, Thailand
Sudarat Nidsunkid Department of Statistics, Faculty of Science, Kasetsart University, Bangkok, Thailand
Dusit Chaiprasithikul Department of Statistics, Faculty of Science, Silpakorn University, Nakhon Pathom, Thailand

Keywords:

maximum likelihood estimation, predictive distribution, count data, Metropolis-Hastings algorithm, cross-validation

Abstract

This paper proposes the Bayesian estimation of the zero-inflated discrete Weibull distribution assuming three prior distributions, namely Beta-Uniform-Uniform prior, Beta-Jeffreys’ rule prior, and Beta-Beta-Gamma prior. It is commonly known that there is no compact form for the Bayes estimators. The Bayesian estimate of the model parameters has been performed through the random walk Metropolis algorithm. The Bayes estimates of the unknown parameters and the credible interval construction are established based on the generated samples. Moreover, the maximum likelihood estimation is considered, as well as the confidence interval estimation for the model parameters has been performed through normal approximation. The performance of the Bayes estimators has also been compared with the classical estimators through the Monte Carlo simulation study. Further, the Bayesian prediction of a future observation is proposed under the three prior distributions. The posterior predictive distribution of a future observation cannot be evaluated analytically. To obtain the estimate of future sample, the Metropolis-Hastings algorithm is used. Two datasets have been analyzed to show how the proposed model and the method work in practice.

References

Altun E, Alqifari H, Eliwa MS. A novel approach for zero-inflated count regression model: Zero inflated Poisson generalized-Lindley linear model with applications. AIMS Math. 2023; 8(10): 23272-23290.

Allison PD, Waterman RP. Fixed-effects negative binomial regression models. Sociol Methodol. 2002; 32(1): 247-265.

Ashour SK, Muiftah MSA. Bayesian estimation of the parameters of discrete Weibull type (I) distribution. J Mod Appl Stat Methods. 2019; 18(2): 1-13.

Bekalo DB, Kebede DT. Zero‑inflated models for count data: an application to number of antenatal care service visits. Ann Data Sci 2021; 8(4): 683-708.

Chaiprasithikul D, Duangsaphon M. Bayesian inference of discrete Weibull regression model for excess zero counts. Sci Technol Asia. 2022; 27(4): 152-174.

Chaiprasithikul D, Duangsaphon M. Bayesian inference for the discrete Weibull regression model with type-I right censored data. Thail Stat. 2022; 20(4): 791-811.

Collins K, Waititu A, Wanjoya A. Discrete Weibull and artificial neural network models in modelling over-dispersed count data. Int J Data Sci Anal. 2020; 6(5): 153-162.

CRAN Team, The comprehensive R archive network, 2023. https://cran.r-project.org/.

Dai YH. A perfect example for the BFGS method. Math Program. 2013; 138: 501-530.

Duangsaphon M, Santimalai R, Volodin A. Bayesian estimation and prediction for discrete Weibull distribution. Lobachevskii J Math. 2023; 44(11): 4693-4703.

Duangsaphon M, Sokampang S, Na Bangchang K. Bayesian estimation for median discrete Weibull regression model. AIMS Math. 2024; 9(1): 270-288.

Feng CX. A comparison of zero-inflated and hurdle models for modeling zero-inflated count data. J Stat Distrib Appl. 2021: 8(8): 1-9.

Gardner W, Mulvey EP, Shaw EC. Regression analyses of counts and rates: Poisson, overdispersed Poisson, and negative binomial models. Psychol. Bull. 1995; 118(3): 392-404.

Ghosh JK, Delampady M, Samanta T. An introduction to Bayesian analysis: Theory and methods. New York: Springer; 2006.

Gilks WR, Richardson S, Spiegelhalter D. Markov chain Monte Carlo in practice. Interdisciplinary Statistics. London: Chapman and Hall; 1996.

Haselimashhadi H, Vinciotti V, Yu K. A novel Bayesian regression model for counts with an application to health data. J Appl Stat. 2018; 45(6): 1085-1105.

Hastings WK. Monte Carlo sampling methods using Markov chains and their applications. Biometrika. 1970; 57(1970): 97-109.

Hosmer DW, Lemeshow JS. Applied logistic regression. John Wiley & Sons: New York; 2004.

Irfan M, A. K. Sharma MA. Bayesian estimation and prediction for inverse power Maxwell distribution with applications to tax revenue and health care data. J Mod Appl Stat Methods. 2024; 23(1): 1-28.

Kalktawi HS. Discrete Weibull regression model for count data. PhD [dissertation]. London: Brunel University London; 2017.

Kalktawi HS, Vinciotti V, Yu K. A simple and adaptive dispersion regression model for count data. Entropy. 2018; 20(2): 1-15.

Liu H, Davidson RA, Rosowsky DV, Stedinger JR. Negative binomial regression of electric power outages in hurricanes. J Infrastruct Syst. 2005; 11(4): 258-267.

Nakagawa T, Osaki S. The discrete Weibull distribution. IEEE Trans Reliab. 1975; R-24(5): 300-301.

R Core Team and Contributors Worldwide, Cubature, Adaptive multivariate integration over hypercubes, 2023. https://cran.r-project.org/web/packages/cubature/index.html.

R Core Team and Contributors Worldwide, General-Purpose Optimization, 2022.

R Core Team and Contributors Worldwide, Parametric regression for discrete response, 2016. https://CRAN.R-project.org/package=DWreg.

Sellers KF, Shmueli AG. A flexible regression model for count data. Ann Appl Stat. 2010; 4: 943-961.

Serfling RJ. Approximation theorems of mathematical statistics: New York: JohnWiley & Sons; 1980.

Singh KS, Singh U, Sharma VK. Bayesian estimation and prediction for flexible Weibull model under type-II censoring scheme. J Prob Stat. 2013; 2013(146140): 1-16.

Taveekal P, Rajchanuwong P, Wongwiangjan R, Lerdsuwansri R, Intrakul J, Simmachan T, Wongsai S. Modelling road accident injuries and fatalities in Suratthani province of Thailand using Conway-Maxwell-Poisson Regression. Thail Stat. 2023; 21(3): 569-579.

Unhapipat S, Tiensuwan M, Pal N. Bayesian predictive inference for zero-inflated Poisson (ZIP) distribution with applications. Am J Math Manag Sci. 2018; 37(1) 66-79.

Wang S, Chen W, Chen M, Zhou Y. Maximum likelihood estimation of the parameters of the inverse Gaussian distribution using maximum rank set sampling with unequal samples. Math Popul Stud. 2023; 30: 1-21.