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The Large Arcsine Exponential Dispersion Model—Properties and Applications to Count Data and Insurance Risk

Author

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  • Shaul K. Bar-Lev

    (Faculty of Industrial Engineering and Technology Management, HIT—Holon Institute of Technology, Holon 5810201, Israel)

  • Ad Ridder

    (School of Business and Economics, Vrije University, 1081 HV Amsterdam, The Netherlands)

Abstract

The large arcsine exponential dispersion model (LAEDM) is a class of three-parameter distributions on the non-negative integers. These distributions show the specific characteristics of being leptokurtic, zero-inflated, overdispersed, and skewed to the right. Therefore, these distributions are well suited to fit count data with these properties. Furthermore, recent studies in actuarial sciences argue for the consideration of such distributions in the computation of risk factors. In this paper, we provide a thorough analysis of the LAEDM by deriving (a) the mean value parameterization of the LAEDM; (b) exact expressions for its probability mass function at n = 0 , 1 , … ; (c) a simple bound for these probabilities that is sharp for large n ; (d) a simulation algorithm for sampling from LAEDM. We have implemented the LAEDM for statistical modeling of various real count data sets. We assess its fitting performance by comparing it with the performances of traditional counting models. We use a simulation algorithm for computing tail probabilities of the aggregated claim size in an insurance risk model.

Suggested Citation

  • Shaul K. Bar-Lev & Ad Ridder, 2022. "The Large Arcsine Exponential Dispersion Model—Properties and Applications to Count Data and Insurance Risk," Mathematics, MDPI, vol. 10(19), pages 1-25, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3715-:d:938241
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    References listed on IDEAS

    as
    1. Hilbe,Joseph M., 2014. "Modeling Count Data," Cambridge Books, Cambridge University Press, number 9781107611252, January.
    2. Bent Jørgensen & Célestin Kokonendji, 2016. "Discrete dispersion models and their Tweedie asymptotics," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 100(1), pages 43-78, January.
    3. Kunreuther, Howard & Novemsky, Nathan & Kahneman, Daniel, 2001. "Making Low Probabilities Useful," Journal of Risk and Uncertainty, Springer, vol. 23(2), pages 103-120, September.
    4. Rahma Abid & Célestin C. Kokonendji & Afif Masmoudi, 2021. "On Poisson-exponential-Tweedie models for ultra-overdispersed count data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(1), pages 1-23, March.
    5. Shaul K. Bar-Lev & Ad Ridder, 2019. "Monte Carlo Methods for Insurance Risk Computation," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 8(3), pages 1-54, November.
    6. Yaser Awad & Shaul K. Bar-Lev & Udi Makov, 2022. "A New Class of Counting Distributions Embedded in the Lee–Carter Model for Mortality Projections: A Bayesian Approach," Risks, MDPI, vol. 10(6), pages 1-17, May.
    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. Deepesh Bhati & Hassan S. Bakouch, 2019. "A new infinitely divisible discrete distribution with applications to count data modeling," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(6), pages 1401-1416, March.
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