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A New Family of Discrete Distributions with Mathematical Properties, Characterizations, Bayesian and Non-Bayesian Estimation Methods

Author

Listed:
  • Mohamed Aboraya

    (Department of Applied Statistics and Insurance, Faculty of Commerce, Damietta University, Damietta 34519, Egypt)

  • Haitham M. Yousof

    (Department of Statistics, Mathematics and Insurance, Benha University, Banha 13513, Egypt)

  • G.G. Hamedani

    (Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, WI 53201-1881, USA)

  • Mohamed Ibrahim

    (Department of Applied Statistics and Insurance, Faculty of Commerce, Damietta University, Damietta 34519, Egypt)

Abstract

In this work, we propose and study a new family of discrete distributions. Many useful mathematical properties, such as ordinary moments, moment generating function, cumulant generating function, probability generating function, central moment, and dispersion index are derived. Some special discrete versions are presented. A certain special case is discussed graphically and numerically. The hazard rate function of the new class can be “decreasing”, “upside down”, “increasing”, and “decreasing-constant-increasing (U-shape)”. Some useful characterization results based on the conditional expectation of certain function of the random variable and in terms of the hazard function are derived and presented. Bayesian and non-Bayesian methods of estimation are considered. The Bayesian estimation procedure under the squared error loss function is discussed. Markov chain Monte Carlo simulation studies for comparing non-Bayesian and Bayesian estimations are performed using the Gibbs sampler and Metropolis–Hastings algorithm. Four applications to real data sets are employed for comparing the Bayesian and non-Bayesian methods. The importance and flexibility of the new discrete class is illustrated by means of four real data applications.

Suggested Citation

  • Mohamed Aboraya & Haitham M. Yousof & G.G. Hamedani & Mohamed Ibrahim, 2020. "A New Family of Discrete Distributions with Mathematical Properties, Characterizations, Bayesian and Non-Bayesian Estimation Methods," Mathematics, MDPI, vol. 8(10), pages 1-25, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1648-:d:418504
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    References listed on IDEAS

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    1. Bebbington, Mark & Lai, Chin-Diew & Wellington, Morgan & Zitikis, RiÄ ardas, 2012. "The discrete additive Weibull distribution: A bathtub-shaped hazard for discontinuous failure data," Reliability Engineering and System Safety, Elsevier, vol. 106(C), pages 37-44.
    2. Mahdi Rasekhi & Mohammad Mehdi Saber & Haitham M. Yousof, 2020. "Bayesian and classical inference of reliability in multicomponent stress-strength under the generalized logistic model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(21), pages 5114-5125, September.
    3. Li Cai, 2010. "Metropolis-Hastings Robbins-Monro Algorithm for Confirmatory Item Factor Analysis," Journal of Educational and Behavioral Statistics, , vol. 35(3), pages 307-335, June.
    4. M. El-Morshedy & M. S. Eliwa & H. Nagy, 2020. "A new two-parameter exponentiated discrete Lindley distribution: properties, estimation and applications," Journal of Applied Statistics, Taylor & Francis Journals, vol. 47(2), pages 354-375, January.
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    Cited by:

    1. Mohamed Ibrahim & M. Masoom Ali & Haitham M. Yousof, 2023. "The Discrete Analogue of the Weibull G Family: Properties, Different Applications, Bayesian and Non-Bayesian Estimation Methods," Annals of Data Science, Springer, vol. 10(4), pages 1069-1106, August.
    2. Manal M. Yousef & Amal S. Hassan & Abdullah H. Al-Nefaie & Ehab M. Almetwally & Hisham M. Almongy, 2022. "Bayesian Estimation Using MCMC Method of System Reliability for Inverted Topp–Leone Distribution Based on Ranked Set Sampling," Mathematics, MDPI, vol. 10(17), pages 1-26, August.

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