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The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality

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  • Alexey Kudryavtsev

    (Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia)

  • Oleg Shestakov

    (Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia
    Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia)

Abstract

When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support.

Suggested Citation

  • Alexey Kudryavtsev & Oleg Shestakov, 2022. "The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality," Mathematics, MDPI, vol. 10(4), pages 1-17, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:4:p:619-:d:751475
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    References listed on IDEAS

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    1. N. J. Hassan & J. Mahdi Hadad & A. Hawad Nasar, 2020. "Bayesian Shrinkage Estimator of Burr XII Distribution," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2020, pages 1-6, June.
    2. Yi Zhou & Hongqing Zhu, 2018. "Image Segmentation Using a Trimmed Likelihood Estimator in the Asymmetric Mixture Model Based on Generalized Gamma and Gaussian Distributions," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-17, January.
    3. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
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