### Bayesian and maximum likelihood inference approaches for the discrete generalized Sibuya distribution with censored data

#### Abstract

#### References

Abramowitz, M. and Stegun I. A. (eds.) (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55. Dover Publications, New York.

Akaike, H. (1974). A new look at statistical model identification. IEEE Transactions on Automatic Control, 19:716-723.

Allison, P. D. (1982). Discrete-time methods for the analysis of event histories. Sociological Methodology, 13:61-98.

Anderson, D. and Burnham, K. (2002). Model selection and multi-model inference. 2nd edition. Springer-Verlag, New York.

Bouzar, N. (2008). The semi-Sibuya distribution. Annals of the Institute of Statistical Mathematics, 60(2):459-464.

Buddana, A. and Kozubowski, T. J. (2014). Discrete Pareto distributions. Stochastics and Quality Control, 29(2):143-156.

Cardial, M.R.P., Fachini-Gomes, J.B., and Nakano, E. Y. (2020). Exponentiated discrete Weibull distribution for censored data. Brazilian Journal of Biometrics, 38(1), 35-56.

Chib, S. and Greenberg, E. (1995). Understanding the Metropolis-Hastings algorithm. The American Statistician, 49(4), 327-335.

Christoph, G. and Schreiber, K. (2000). Scaled Sibuya distribution and discrete self-decomposability. Statistics & Probability Letters, 48(2), 181-187.

Devroye, L. (1993). A triptych of discrete distributions related to the stable law. Statistics & Probability Letters, 18(5), 349-351.

Dey, D. K., Chen, M. H., and Chang, H. (1997). Bayesian approach for nonlinear random effects models. Biometrics, 53(4), 1239-1252.

Drevon, D., Fursa, S. R., and Malcolm, A. L. (2017). Intercoder reliability and validity of WebPlotDigitizer in extracting graphed data. Behavior Modification, 41(2), 323-339.

Eldeeb, A. S., Ahsan-ul-Haq, M., Eliwa, M. S., and Cell, Q. E. (2022). A discrete Ramos-Louzada distribution for asymmetric and over-dispersed data with leptokurtic-shaped:Properties and various estimation techniques with inference. AIMS Mathematics, 7(2),1726-1741.

Freitas, B. C. L., Oliveira-Peres, M. V., Achcar, J. A., and Martinez, E. Z. (2021). Classical and Bayesian inference approaches for the exponentiated discrete Weibull model with censored data and a cure fraction. Pakistan Journal of Statistics and Operation Research, 17(2), 467-481.

Gallardo, D. I., Gomez, H. W., and Bolfarine, H. (2017). A new cure rate model based on the Yule-Simon distribution with application to a melanoma data set. Journal of Applied Statistics, 44(7):1153-1164.

Geisser, S., and Eddy, W. F. (1979). A predictive approach to model selection. Journal of the American Statistical Association, 74(365), 153-160.

Gelfand, A. E. and Smith, A. F. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85(410), 398-409.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis, 3th Edition. Chapman and Hall/CRC.

Geweke J (1992). Bayesian Statistics, volume 4, chapter Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. Bayesian Statistics, 4:641-649.

Graf, M. and Nedyalkova, D. (2015). GB2: Generalized Beta Distribution of the Second Kind: Properties, Likelihood, Estimation. R Package Version 2.1

Gupta, R. C. and Huang, J. (2017). The Weibull-Conway-Maxwell-Poisson distribution to analyze survival data. Journal of Computational and Applied Mathematics, 311:171-182.

Henningsen, A. and Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics, 26(3):443-458.

Herzog, W., Schellberg, D., and Deter, H. C. (1997). First recovery in anorexia nervosa patients in the long-term course: a discrete-time survival analysis. Journal of Consulting and Clinical Psychology, 65(1):169-177.

Jayakumar, K. and Babu, M. G. (2018). Discrete Weibull geometric distribution and its properties. Communications in Statistics - Theory and Methods, 47(7):1767-1783.

Klein, J. P. and Moeschberger, M. L. (2003). Survival analysis: techniques for censored and truncated data. New York: Springer.

Kozubowski, T. J. and Podgorski, K. (2018). A generalized Sibuya distribution. Annals of the Institute of Statistical Mathematics, 70(4):855-887.

Kruschke, J. (2014). Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Academic Press.

Letac, G. (2019). Is the Sibuya distribution a progeny?. Journal of Applied Probability, 56(1):52-56.

Maity, A., Williams, P. L., Ryan, L., Missmer, S. A., Coull, B. A., and Hauser, R. (2014). Analysis of in vitro fertilization data with multiple outcomes using discrete time-to-event analysis. Statistics in Medicine, 33(10):17381749.

Maity, A. K., Basu, S., and Ghosh, S. (2021). Bayesian criterionbased variable selection. Journal of the Royal Statistical Society: Series C (Applied Statistics), 70(4):835857.

Martin, A. D. and Quinn, K. M. (2006). Applied Bayesian inference in R using MCMCpack. R News, 6(1):2-7.

Martin, A. D., Quinn, K. M., and Park, J. H. (2011). MCMCpack: Markov chain Monte Carlo in R. Journal of Statistical Software, 42(9):1-21.

Nakagawa, T. and Osaki, S. (1975). The discrete Weibull distribution. IEEE Transactions on Reliability, 24(5):300-301.

Nekoukhou, V. and Bidram, H. (2015). The exponentiated discrete Weibull distribution. SORT Statistics and Operations Research Transactions, 39(1): 127-146.

Northrup, T. F., Stotts, A. L., Green, C., Potter, J. S., Marino, E. N., Walker, R., Weiss, R. D., and Trivedi, M. (2015). Opioid withdrawal, craving, and use during and after outpatient buprenorphine stabilization and taper: a discrete survival and growth mixture model. Addictive Behaviors, 41:20-28.

Oddy, W. H., Li, J., Landsborough, L., Kendall, G. E., Henderson, S., and Downie, J. (2006). The association of maternal overweight and obesity with breastfeeding duration. The Journal of Pediatrics, 149(2):185-191.

Oehlert, G. W. (1992). A note on the delta method. The American Statistician, 46(1):27-29.

Ramos, P. L., Guzman, D. C., Mota, A. L., Rodrigues, F. A., and Louzada, F. (2020). Sampling with censored data: a practical guide. arXiv preprint, arXiv:2011.08417.

Rohatgi, A. (2020). WebPlotDigitizer (Version 4.4) [Computer software]. Retrieved from https://automeris.io/WebPlotDigitizer/

Scheike, T. H. and Jensen, T. K. (1997). A discrete survival model with random effects: an application to time to pregnancy. Biometrics, 53(1):318-329.

Sibuya, M. (1979). Generalized hypergeometric, digamma and trigamma distributions. Annals of the Institute of Statistical Mathematics, 31(3):373-390.

Simon, H. A. (1955). On a class of skew distribution functions. Biometrika, 42(3-4):425-440.

Singer, J. D. and Willett, J. B. (1993). Its about time: Using discrete-time survival analysis to study duration and the timing of events. Journal of Educational Statistics, 18(2):155-195.

Tutz, G. and Schmid, M. (2016). Modeling discrete time-to-event data. New York: Springer.

Vanegas, J. C., Chavarro, J. E., Williams, P. L., Ford, J. B., Toth, T. L., Hauser, R., and Gaskins, A. J. (2017). Discrete survival model analysis of a couples smoking pattern and outcomes of assisted reproduction. Fertility Research and Practice, 3(1):5.

Yule, G. U. (1924). A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S. Philosophical Transactions of the Royal Society B, 213(402-410):21-87.

Full Text: pdf