# Gamma Zero-Truncated Poisson Distribution with the Minimum Compounded Function

## Keywords:

Compounding, gamma distribution, Zero-truncated Poisson distributions## Abstract

The gamma zero-truncated Poisson (GZTP) distribution is introduced in this work as a novel lifetime distribution created by compounding the gamma and zero-truncated Poisson distributions with the minimum function. The proposed distribution’s features are examined, including proofs of its probability density function and cumulative distribution function and formulas for its survival function, hazard function, moment, mean, variance, and quantile. The shape of the GZTP distribution’s hazard function is flexible and can be increasing, decreasing, or unimodal. The estimation process utilizes maximum likelihood. Asymptotic properties of maximum likelihood estimators are studied, and simulations are used to test how well parameter estimation works.

## References

Adamidis K, Dimitrakopoulou T, Loukas S. On an extension of the exponential-geometric distribution. Stat Prob Lett 2005; 73: 259-269.

Adamidis K, Loukas S. A lifetime distribution with decreasing failure rate. Stat Prob Lett. 1998; 39(1): 35-42.

Alkarni S, Oraby A. A compound class of Poisson and lifetime distributions. J Stat Appl Prob. 2012; 1(1): 45-51.

Barreto-Souza W, de Morais AL, Cordeiro GM. The Weibull-geometric distribution. J Stat Comput Simul. 2011; 81(5): 645-657.

Barreto-Souza W, Silva RB. A likelihood ratio test to discriminate exponential-Poisson and gamma distributions. J Stat Comput Simul 2015; 85: 802-823.

Ciumara R, Preda V. The Weibull-Logarithmic distribution in lifetime analysis and its properties. In: Sakalauskas L, Skiadas C, Zavadskas EK, editors. ASMDA-2009: Proceedings of the XIIIth International Conference “Applied Stochastic Models and Data Analysis”; 2009 June 30-July 3; Lithuania. Institute of Mathematics and Information/Vilnius Gediminas Technical University; 2009. pp. 395–399.

Glaser RE. Bathtub and related failure rate characterizations. J Am Stat Assoc. 1980; 75(371): 667-672.

Gui W, Zhang S, Lu X. The Lindley-Poisson distribution in lifetime analysis and its properties. Hacet J Math Stat. 2014; 43(2): 1063-1077.

Hemmati F, Khorram E, Rezakhah S. A new three-parameter ageing distribution. J Stat Plan Inference. 2011; 141(7): 2266-2275.

Henningsen A, Toomet O. maxLik: A package for maximum likelihood estimation in R. Comput Stat. 2011; 26(3): 443-458.

Kuş C. A new lifetime distribution. Comput Stat Data Anal. 2007; 51(9): 4497-4509.

Lee ET, Wang J. Statistical methods for survival data analysis. 3rd ed. John Wiley & Sons; 2003.

Louzada F, Luiz Ramos P, Henrique Ferreira P. Exponential-Poisson distribution: estimation and applications to rainfall and aircraft data with zero occurrence. Commun Stat Simul Comput. 2020; 49(4): 1024-1043.

Louzada F, Ramos PL, Perdoná GSC. Different estimation procedures for the parameters of the extended exponential geometric distribution for medical data. Comput Math Methods Med 2016; 2016:8727951.

Lu W, Shi D. A new compounding life distribution: The Weibull-Poisson distribution. J Appl Stat. 2012; 39(1): 21-38.

Tahmasbi R, Rezaei S. A two-parameter lifetime distribution with decreasing failure rate. Comput Stat Data Anal. 2008; 52(8): 3889-3901.

Xu B, Guo Y, Zhu N. The parameter bayesian estimation of two-parameter exponential-poisson distribution and its optimal property. J Interdiscip Math 2016; 19: 697–707.

Zakerzadeh H, Mahmoudi E. A new two parameter lifetime distribution: model and properties. arXiv.1204.4248v1 [Preprint]. 2012. [cited 2021 December 25]. Available from: https://doi.org/

48550/arXiv.1204.4248.

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*Thailand Statistician*,

*21*(4), 863–886. Retrieved from https://ph02.tci-thaijo.org/index.php/thaistat/article/view/251065

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