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Theory and Applications of the Unit Gamma/Gompertz Distribution

Author

Listed:
  • Rashad A. R. Bantan

    (Department of Marine Geology, Faculty of Marine Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia)

  • Farrukh Jamal

    (Department of Statistics, The Islamia University of Bahawalpur, Punjab 63100, Pakistan)

  • Christophe Chesneau

    (Department of Mathematics, Université de Caen, LMNO, Campus II, Science 3, 14032 Caen, France)

  • Mohammed Elgarhy

    (The Higher Institute of Commercial Sciences, Al Mahalla Al Kubra, Algarbia 31951, Egypt)

Abstract

Unit distributions are commonly used in probability and statistics to describe useful quantities with values between 0 and 1, such as proportions, probabilities, and percentages. Some unit distributions are defined in a natural analytical manner, and the others are derived through the transformation of an existing distribution defined in a greater domain. In this article, we introduce the unit gamma/Gompertz distribution, founded on the inverse-exponential scheme and the gamma/Gompertz distribution. The gamma/Gompertz distribution is known to be a very flexible three-parameter lifetime distribution, and we aim to transpose this flexibility to the unit interval. First, we check this aspect with the analytical behavior of the primary functions. It is shown that the probability density function can be increasing, decreasing, “increasing-decreasing” and “decreasing-increasing”, with pliant asymmetric properties. On the other hand, the hazard rate function has monotonically increasing, decreasing, or constant shapes. We complete the theoretical part with some propositions on stochastic ordering, moments, quantiles, and the reliability coefficient. Practically, to estimate the model parameters from unit data, the maximum likelihood method is used. We present some simulation results to evaluate this method. Two applications using real data sets, one on trade shares and the other on flood levels, demonstrate the importance of the new model when compared to other unit models.

Suggested Citation

  • Rashad A. R. Bantan & Farrukh Jamal & Christophe Chesneau & Mohammed Elgarhy, 2021. "Theory and Applications of the Unit Gamma/Gompertz Distribution," Mathematics, MDPI, vol. 9(16), pages 1-22, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1850-:d:608851
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    References listed on IDEAS

    as
    1. J. Mazucheli & A. F. B. Menezes & L. B. Fernandes & R. P. de Oliveira & M. E. Ghitany, 2020. "The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates," Journal of Applied Statistics, Taylor & Francis Journals, vol. 47(6), pages 954-974, April.
    2. Gómez-Déniz, Emilio & Sordo, Miguel A. & Calderín-Ojeda, Enrique, 2014. "The Log–Lindley distribution as an alternative to the beta regression model with applications in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 49-57.
    3. Emrah Altun, 2021. "The log-weighted exponential regression model: alternative to the beta regression model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(10), pages 2306-2321, May.
    4. Albert C. Bemmaor & Nicolas Glady, 2012. "Modeling Purchasing Behavior with Sudden "Death": A Flexible Customer Lifetime Model," Management Science, INFORMS, vol. 58(5), pages 1012-1021, May.
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