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Limit Distributions for the Estimates of the Digamma Distribution Parameters Constructed from a Random Size Sample

Author

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  • Alexey Kudryavtsev

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

  • Oleg Shestakov

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

Abstract

In this paper, we study a new type of distribution that generalizes distributions from the gamma and beta classes that are widely used in applications. The estimators for the parameters of the digamma distribution obtained by the method of logarithmic cumulants are considered. Based on the previously proved asymptotic normality of the estimators for the characteristic index and the shape and scale parameters of the digamma distribution constructed from a fixed-size sample, we obtain a statement about the convergence of these estimators to the scale mixtures of the normal law in the case of a random sample size. Using this result, asymptotic confidence intervals for the estimated parameters are constructed. A number of examples of the limit laws for sample sizes with special forms of negative binomial distributions are given. The results of this paper can be widely used in the study of probabilistic models based on continuous distributions with an unbounded non-negative support.

Suggested Citation

  • Alexey Kudryavtsev & Oleg Shestakov, 2023. "Limit Distributions for the Estimates of the Digamma Distribution Parameters Constructed from a Random Size Sample," Mathematics, MDPI, vol. 11(8), pages 1-13, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1778-:d:1118637
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    References listed on IDEAS

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    1. Deepesh Bhati & Ishfaq S. Ahmed, 2021. "On uniform-negative binomial distribution including Gauss hypergeometric function and its application in count regression modeling," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(13), pages 3106-3122, July.
    2. 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.
    3. Alexey Kudryavtsev & Oleg Shestakov, 2021. "Asymptotically Normal Estimators for the Parameters of the Gamma-Exponential Distribution," Mathematics, MDPI, vol. 9(3), pages 1-13, January.
    4. Shuhan Liu & Wenhao Gui, 2020. "Estimating the Parameters of the Two-Parameter Rayleigh Distribution Based on Adaptive Type II Progressive Hybrid Censored Data with Competing Risks," Mathematics, MDPI, vol. 8(10), pages 1-16, October.
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    1. Alexey Kudryavtsev & Oleg Shestakov, 2023. "Estimates of the Convergence Rate in the Generalized Rényi Theorem with a Structural Digamma Distribution Using Zeta Metrics," Mathematics, MDPI, vol. 11(21), pages 1-10, October.

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