The discrete Weibull model can be adapted to capture different levels of dispersion in the count data. This paper takes into account the direct relationship between explanatory variables and the median of discrete Weibull response variable. Additionally, it provides the Bayesian estimate of the discrete Weibull regression model using the random walk Metropolis algorithm. The prior distributions of the coefficient predictors were carried out based on the uniform non-informative, normal and Laplace distributions. The performance of the Bayes estimators was also compared with the maximum likelihood estimator in terms of the mean square error and the coverage probability through the Monte Carlo simulation study. Meanwhile, a real data set was analyzed to show how the proposed model and the methods work in practice.
Citation: Monthira Duangsaphon, Sukit Sokampang, Kannat Na Bangchang. Bayesian estimation for median discrete Weibull regression model[J]. AIMS Mathematics, 2024, 9(1): 270-288. doi: 10.3934/math.2024016
The discrete Weibull model can be adapted to capture different levels of dispersion in the count data. This paper takes into account the direct relationship between explanatory variables and the median of discrete Weibull response variable. Additionally, it provides the Bayesian estimate of the discrete Weibull regression model using the random walk Metropolis algorithm. The prior distributions of the coefficient predictors were carried out based on the uniform non-informative, normal and Laplace distributions. The performance of the Bayes estimators was also compared with the maximum likelihood estimator in terms of the mean square error and the coverage probability through the Monte Carlo simulation study. Meanwhile, a real data set was analyzed to show how the proposed model and the methods work in practice.
| [1] | A. C. Cameron, P. K. Trivedi, Regression analysis of count data, Cambridge: Cambridge University Press, 2013. https://doi.org/10.1017/CBO9781139013567 |
| [2] |
K. F. Sellers, G. Shmueli, A flexible regression model for count data, Ann. Appl. Stat., 4 (2010), 943–961. https://doi.org/10.1214/09-AOAS306 doi: 10.1214/09-AOAS306
|
| [3] |
D. Lambert, Zero-inflated Poisson regression, with an application to defects in manufacturing, Technometrics, 34 (1992), 1–14. https://doi.org/10.2307/1269547 doi: 10.2307/1269547
|
| [4] |
D. B. Hall, Zero-inflated Poisson and binomial regression with random effects: A case study, Biometrics, 56 (2000), 1030–1039. https://doi.org/10.1111/j.0006-341X.2000.01030.x doi: 10.1111/j.0006-341X.2000.01030.x
|
| [5] |
K. K. W. Yau, K. Wang, A. H. Lee, Zero-inflated negative binomial mixed regression modeling of over-dispersed count data with extra zeros, Biomretical J., 45 (2003), 437–452. https://doi.org/10.1002/bimj.200390024 doi: 10.1002/bimj.200390024
|
| [6] |
T. Nakagawa, S. Osaki, The discrete Weibull distribution, IEEE T. Reliab., 24 (1975), 300–301. https://doi.org/10.1109/TR.1975.5214915 doi: 10.1109/TR.1975.5214915
|
| [7] |
L. Dang, X. He, D. Tang, Y. Li, T. Wang, A fatigue life prediction approach for laser-directed energy deposition titanium alloys by using support vector regression based on pore-induced failures, Int. J. Fatigue, 159 (2022), 106748. https://doi.org/10.1016/j.ijfatigue.2022.106748 doi: 10.1016/j.ijfatigue.2022.106748
|
| [8] |
J. C. He, S. P. Zhu, C. Luo, X. Niu, Q. Wang, Size effect in fatigue modelling of defective materials: Application of the calibrated weakest-link theory, Int. J. Fatigue, 165 (2022), 10721. https://doi.org/10.1016/j.ijfatigue.2022.107213 doi: 10.1016/j.ijfatigue.2022.107213
|
| [9] | H. S. Kalktawi, Discrete Weibull regression model for count data, PhD. thesis, London: Brunel University London, 2017. |
| [10] |
J. D. Englehardt, R. Li, The discrete Weibull distribution: An alternative for correlated counts with confirmation for microbial counts in water, Risk Anal., 31 (2011), 370–381. https://doi.org/10.1111/j.1539-6924.2010.01520.x doi: 10.1111/j.1539-6924.2010.01520.x
|
| [11] |
H. S. Klakattawi, V. Vinciotti, K. Yu, A simple and adaptive dispersion regression model for count data, Entropy, 20 (2018), 142. https://doi.org/10.3390/e20020142 doi: 10.3390/e20020142
|
| [12] |
A. Peluso, V. Vinciotti, Discrete weibull generalised additive model: An application to count fertility data, J. Roy. Stat. Soc. C-Appl., 2018. https://doi.org/10.1111/rssc.12311 doi: 10.1111/rssc.12311
|
| [13] |
S. Wang, W. Chen, M. Chen, Y. Zhou, Maximum likelihood estimation of the parameters of the inverse Gaussian distribution using maximum rank set sampling with unequal samples, Math. Popul. Stud., 30 (2023), 1–21. https://doi.org/10.1080/08898480.2021.1996822 doi: 10.1080/08898480.2021.1996822
|
| [14] |
H. Haselimashhadi, V. Vinciotti, K. Yu, A novel Bayesian regression model for counts with an application to health data, J. Appl. Stat., 45 (2018), 1085–1105. https://doi.org/10.1080/02664763.2017.1342782 doi: 10.1080/02664763.2017.1342782
|
| [15] |
A. Gelman, A. Jakulin, M. G. Pittau, Y. Su, A weakly informative default prior distribution for logistic and other regression models, Ann. Appl. Stat., 2 (2008), 1360–1383. https://doi.org/10.1214/08-AOAS191 doi: 10.1214/08-AOAS191
|
| [16] |
S. Fu, Hierarchical Bayesian LASSO for a negative binomial regression, J. Stat. Comput. Sim., 86 (2016), 2182–2203. https://doi.org/10.1080/00949655.2015.1106541 doi: 10.1080/00949655.2015.1106541
|
| [17] |
C. Chanialidis, L. Evers, T. Neocleous, A. Nobile, Efficient Bayesian inference for COM-Poisson regression models, Stat. Comput., 28 (2018), 595–608. https://doi.org/10.1007/s11222-017-9750-x doi: 10.1007/s11222-017-9750-x
|
| [18] | D. Chaiprasithikul, M. Duangsaphon, Bayesian inference for the discrete Weibull regression model with Type-Ⅰ right censored data, Thail. Statist., 20 (2022), 791–811. |
| [19] | D. Chaiprasithikul, M. Duangsaphon, Bayesian inference of discrete Weibull regression model for excess zero counts, Sci. Tech. Asia, 27 (2022), 152–174. |
| [20] |
A. A. Ahmadini, A. S. Hassan, A. N. Zaky, S. S. Alshqaq, Bayesian inference of dynamic cumulative residual entropy from Pareto Ⅱ distribution with application to COVID-19, AIMS Math., 6 (2021), 2196–2216. https://doi.org/10.3934/math.2021133 doi: 10.3934/math.2021133
|
| [21] |
H. Okasha, M. Nassar, S. A. Dobbah, E-Bayesian estimation of Burr Type XⅡ model based on adaptive Type-Ⅱ progressive hybrid censored data, AIMS Math., 6 (2021), 4173–4196. https://doi.org/10.3934/math.2021247 doi: 10.3934/math.2021247
|
| [22] |
H. Sato, M. Ikota, A. Sugimoto, H. Masuda, A new defect distribution metrology with a consistent discrete exponential formula and its applications, IEEE T. Semiconductor M., 12 (1999), 409–418. https://doi.org/10.1109/66.806118 doi: 10.1109/66.806118
|
| [23] |
D. Roy, Discrete rayleigh distribution, IEEE T. Reliab., 53 (2014), 255–260. https://doi.org/10.1109/TR.2004.829161 doi: 10.1109/TR.2004.829161
|
| [24] | A. Barbiero, DiscreteWeibull: Discrete Weibull distributions (Type 1 and 3), 2015. Available from: http://CRAN.R-project.org/package = DiscreteWeibull.Rpackageversion1.0.1. |
| [25] | R. J. Serfling, Approximation theorems of mathematical statistics, New York: John Wiley & Sons, 1980. https://doi.org/10.1002/9780470316481 |
| [26] |
W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), 97–109. https://doi.org/10.1093/biomet/57.1.97 doi: 10.1093/biomet/57.1.97
|
| [27] | W. R. Gilks, S. Richardson, D. Spiegelhalter, Markov chain Monte Carlo in practice, London: Chapman and Hall, 1996. https://doi.org/10.1201/b14835 |
| [28] | D. W. Hosmer, S. Lemeshow, Applied logistic regression, New York: John Wiley & Sons, 2004. https://doi.org/10.1002/0470011815.b2a10029 |
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Monthira Duangsaphon, Sukit Sokampang, Kannat Na Bangchang. Bayesian estimation for median discrete Weibull regression model[J]. AIMS Mathematics, 2024, 9(1): 270-288. doi: 10.3934/math.2024016
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