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The Discrete Analogue of the Weibull G Family: Properties, Different Applications, Bayesian and Non-Bayesian Estimation Methods

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  • Mohamed Ibrahim

    (Damietta University)

  • M. Masoom Ali

    (Ball State University)

  • Haitham M. Yousof

    (Benha University)

Abstract

In this work, we propose and study a new discrete analogue of the Weibull class. Many useful properties such as ordinary moments; moment generating function; cumulant generating function; probability generating function; central moments and dispersion index are derived. Two special discrete versions are discussed theoretically, graphically, and numerically. The hazard rate function of the new family can be "upside down", "increasing", "decreasing", "constant", "J-hazard rate function" and "double upside down" and "increasing-constant". Non-Bayesian estimation methods such as the maximum likelihood estimation; Cramér-von Mises estimation; the ordinary least square estimation and the weighted least square estimation are considered. The Bayesian estimation procedure under the squared error loss function is also presented. The Markov chain Monte Carlo simulations for comparing non-Bayesian and Bayesian estimation are performed using the Gibbs sampler and Metropolis Hastings algorithm. The flexibility of the new family is illustrated by four real datasets. The new family (through two special discrete versions) provided a better fit than sixteen competitive distributions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aodasc:v:10:y:2023:i:4:d:10.1007_s40745-021-00327-y
    DOI: 10.1007/s40745-021-00327-y
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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. Mahmoud M. Mansour & Mohamed Ibrahim & Khaoula Aidi & Nadeem Shafique Butt & Mir Masoom Ali & Haitham M. Yousof & Mohamed S. Hamed, 2020. "A New Log-Logistic Lifetime Model with Mathematical Properties, Copula, Modified Goodness-of-Fit Test for Validation and Real Data Modeling," Mathematics, MDPI, vol. 8(9), pages 1-20, September.
    3. Mukhtar M. Salah & M. El-Morshedy & M. S. Eliwa & Haitham M. Yousof, 2020. "Expanded Fréchet Model: Mathematical Properties, Copula, Different Estimation Methods, Applications and Validation Testing," Mathematics, MDPI, vol. 8(11), pages 1-29, November.
    4. Faton Merovci & Morad Alizadeh & Haitham M. Yousof & G. G. Hamedani, 2017. "The exponentiated transmuted-G family of distributions: Theory and applications," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(21), pages 10800-10822, November.
    5. M. S. Eliwa & M. El-Morshedy, 2019. "Bivariate Gumbel-G Family of Distributions: Statistical Properties, Bayesian and Non-Bayesian Estimation with Application," Annals of Data Science, Springer, vol. 6(1), pages 39-60, March.
    6. 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.
    7. 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.
    8. M. S. Eliwa & Ziyad Ali Alhussain & M. El-Morshedy, 2020. "Discrete Gompertz-G Family of Distributions for Over- and Under-Dispersed Data with Properties, Estimation, and Applications," Mathematics, MDPI, vol. 8(3), pages 1-26, March.
    9. Mohammed H. AbuJarad & Athar Ali Khan & Mundher A. Khaleel & Eman S. A. AbuJarad & Ali H. AbuJarad & Pelumi E. Oguntunde, 2020. "Bayesian Reliability Analysis of Marshall and Olkin Model," Annals of Data Science, Springer, vol. 7(3), pages 461-489, September.
    10. James M. Tien, 2017. "Internet of Things, Real-Time Decision Making, and Artificial Intelligence," Annals of Data Science, Springer, vol. 4(2), pages 149-178, June.
    11. Hafida Goual & Haitham M. Yousof, 2020. "Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test," Journal of Applied Statistics, Taylor & Francis Journals, vol. 47(3), pages 393-423, February.
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    Cited by:

    1. Fanqun Li & Shanran Wei & Mingtao Zhao, 2023. "Bayesian Estimation of a New Pareto-Type Distribution Based on Mixed Gibbs Sampling Algorithm," Mathematics, MDPI, vol. 12(1), pages 1-13, December.

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