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  • 1 Persian Gulf University, Bushehr, Iran
  • | 2 Fasa University, Fasa, Iran
  • | 3 Marquette University, Milwaukee, USA
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Abstract

In this paper, we introduce a new three-parameter generalized version of the Gompertz model called the odd log-logistic Gompertz (OLLGo) distribution. It includes some well-known lifetime distributions such as Gompertz (Go) and odd log-logistic exponential (OLLE) as special sub-models. This new distribution is quite flexible and can be used effectively in modeling survival data and reliability problems. It can have a decreasing, increasing and bathtub-shaped failure rate function depending on its parameters. Some mathematical properties of the new distribution, such as closed-form expressions for the density, cumulative distribution, hazard rate function, the kth order moment, moment generating function and the quantile measure are provided. We discuss maximum likelihood estimation of the OLLGo parameters as well as three other estimation methods from one observed sample. The flexibility and usefulness of the new distribution is illustrated by means of application to a real data set.

  • [1]

    Aarset, M. V., How to identify a bathtub hazard rate, IEEE Transactions on Reliability, R-36 (1) (1987), 106108.

  • [2]

    Akinsete, A., Famoye, F. and Lee, C., The beta-Pareto distribution, Statistics, 42 (6) (2008), 547563.

  • [3]

    Bemmaor, A. C. and Glady, N., Modeling purchasing behavior with sudden “death”: A flexible customer lifetime model, Management Science 58 (5) (2012), 10121021.

    • Search Google Scholar
    • Export Citation
  • [4]

    Benkhelifa, L., The Marshall-Olkin extended generalized Gompertz distribution, arXiv preprint arXiv:1603.08242, 2016.

  • [5]

    Brown, K. and Forbes, W., A mathematical model of aging processes, Journal of Gerontology, 29 (1) (1974), 4651.

  • [6]

    Cintra, R. J., Râgo, L. C., Cordeiro, G. M. and Nascimento, A. D. C., Beta generalized normal distribution with an application for SAR image processing, Statistics, 48 (2) (2012), 279294.

    • Search Google Scholar
    • Export Citation
  • [7]

    Cooray, K., Generalization of the Weibull distribution: the odd Weibull family, Statistical Modelling, 6 (3) (2006), 265277.

  • [8]

    Cordeiro, G. M. and Nadarajah, S., Closed-form expressions for moments of a class of beta generalized distributions, Brazilian Journal of Probability and Statistics, 25 (1) (2011), 1433.

    • Search Google Scholar
    • Export Citation
  • [9]

    Cox, D. R. and Hinkley, D. V., Theoretical Statistics, Chapman and Hall, London, 1974.

  • [10]

    Economos, A. C., Rate of aging, rate of dying and the mechanism of mortality, Archives of Gerontology and Geriatrics, 1 (1) (1982), 4651.

    • Search Google Scholar
    • Export Citation
  • [11]

    El-Gohary, A., Alshamrani, A. and Al-Otaibi, A. N., The generalized Gompertz distribution, Applied Mathematical Modelling, 37 (1–2) (2013), 1324.

    • Search Google Scholar
    • Export Citation
  • [12]

    Eugene, N., Lee, C. and Famoye, F., Beta-normal distribution and its applications, Communications in Statistics — Theory and Methods, 31 (4) (2002), 497512.

    • Search Google Scholar
    • Export Citation
  • [13]

    Glänzel, W., A characterization theorem based on truncated moments and its application to some distribution families, Mathematical Statistics and Probability Theory (Bad Tatzmannsdorf, 1986), Vol. B, Reidel, Dordrecht, 1987, 7584.

    • Search Google Scholar
    • Export Citation
  • [14]

    Glänzel, W., Some consequences of a characterization theorem based on truncated moments, Statistics: A Journal of Theoretical and Applied Statistics, 21 (4) (1990), 613618.

    • Search Google Scholar
    • Export Citation
  • [15]

    Gleaton, J. U. and Lynch, J. D., Properties of generalized log-logistic families of lifetime distributions, Journal of Probability and Statistical Science, 4 (1) (2006), 5164.

    • Search Google Scholar
    • Export Citation
  • [16]

    Gleaton, J. U. and Rahman, M. M., Asymptotic properties of MLE's for distributions generated from a 2-parameter Weibull distribution by a generalized log-logistic transformation, Journal of Probability and Statistical Science, 8 (2010), 199214.

    • Search Google Scholar
    • Export Citation
  • [17]

    Gleaton, J. U. and Rahman, M. M., Asymptotic properties of MLE's for distributions generated from a 2-parameter inverse Gaussian distribution by a generalized log-logistic transformation, Journal of Probability and Statistical Science, 12 (2014), 8599.

    • Search Google Scholar
    • Export Citation
  • [18]

    Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, Edited by Alan Jeffrey and Daniel Zwillinger, Academic Press, 7th edition, 2007.

    • Search Google Scholar
    • Export Citation
  • [19]

    Gupta, R. C. and Gupta, R. D., Proportional reversed hazard rate model and its applications, Journal of Statistical Planning and Inference, 137 (11) (2007), 35253536.

    • Search Google Scholar
    • Export Citation
  • [20]

    Gupta, R. D. and Kundu, D., Generalized exponential distributions, Australian & New Zealand Journal of Statistics, 41 (2) (1999), 173188.

    • Search Google Scholar
    • Export Citation
  • [21]

    Hamedani, G. G., On certain generalized gamma convolution distributions II, Technical Report No. 484, 2013, MSCS, Marquette University.

    • Search Google Scholar
    • Export Citation
  • [22]

    Jafari, A. A., Tahmasebi, S. and Alizadeh, M., The Beta-Gompertz Distribution, Revista Colombiana de Estadística, 37 (1) (2014), 139156.

    • Search Google Scholar
    • Export Citation
  • [23]

    Johnson, N. L., Kotz, S. and Balakrishnan, N., Continuous Univariate Distributions, volume 2, John Wiley & Sons, New York, second edition, 1995.

    • Search Google Scholar
    • Export Citation
  • [24]

    Kenney, J. F. and Keeping, E., Mathematics of Statistics, D. Van Nostrand Company, 1962.

  • [25]

    Moors, J. J. A., A quantile alternative for kurtosis, Journal of the Royal Statistical Society. Series D (The Statistician), 37 (1) (1988), 2532.

    • Search Google Scholar
    • Export Citation
  • [26]

    Nadarajah, S. and Kotz, S., The beta Gumbel distribution, Mathematical Problems in Engineering, 2004 (4) (2004), 323332.

  • [27]

    Nadarajah, S. and Kotz, S., The beta exponential distribution, Reliability Engineering & System Safety, 91 (6) (2006), 689697.

  • [28]

    Ohishi, K., Okamura, H. and Dohi, T., Gompertz software reliability model: estimation algorithm and empirical validation, Journal of Systems and software, 82 (3) (2009), 535543.

    • Search Google Scholar
    • Export Citation
  • [29]

    Roozegar, R., Tahmasebi, S. and Jafari, A. A., The MCDonald Gompertz distribution: properties and applications, Communication in Statistics: Simulation and Computation, 46 (5) (2017), 33413355.

    • Search Google Scholar
    • Export Citation
  • [30]

    Silva, G. O., Ortega, E. M. and Cordeiro, G. M., The beta modified Weibull distribution, Lifetime Data Analysis, 16 (3) (2010), 409430.

    • Search Google Scholar
    • Export Citation
  • [31]

    Silva, R. C., Sanchez, J. J. D., Lima, F́. P. and Cordeiro, G. M., The Kumaraswamy Gompertz distribution, Journal ofData Science, 13 (2015), 241260.

    • Search Google Scholar
    • Export Citation
  • [32]

    Willemse, W. and Koppelaar, H., Knowledge elicitation of gompertz'law of mortality, Scandinavian Actuarial Journal, 2000 (2) (2000), 168179.

    • Search Google Scholar
    • Export Citation