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A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

Author

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  • Shaul K. Bar-Lev

    (Holon Institute of Technology)

  • Gérard Letac

    (Université Paul Sabatier)

  • Ad Ridder

    (Vrije University)

Abstract

The aim of this paper is to delineate a set of new classes of natural exponential families and their associated exponential dispersion models whose probability distributions are supported on the set of nonnegative integers with positive mass on 0 and 1. The new classes are obtained by considering a specific form of their variance functions. We show that the distributions of all these classes are supported on nonnegative integers, that they are infinitely divisible, and that they are skewed to the right, leptokurtic, over-dispersed, and zero-inflated (relative to the Poisson class). Accordingly, these new classes significantly enrich the set of probability models for modeling zero-inflated and over-dispersed count data. Furthermore, we elaborate on numerical techniques how to compute the distributions of our classes, and apply these to an actual data experiment.

Suggested Citation

  • Shaul K. Bar-Lev & Gérard Letac & Ad Ridder, 2024. "A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(4), pages 679-709, August.
    • Handle: RePEc:spr:aistmt:v:76:y:2024:i:4:d:10.1007_s10463-024-00903-y
    DOI: 10.1007/s10463-024-00903-y
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    References listed on IDEAS

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    1. Letac, Gérard, 2022. "Duality for real and multivariate exponential families," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    2. Rainer Winkelmann, 2008. "Econometric Analysis of Count Data," Springer Books, Springer, edition 0, number 978-3-540-78389-3, March.
    3. Cindy Xin Feng, 2021. "A comparison of zero-inflated and hurdle models for modeling zero-inflated count data," Journal of Statistical Distributions and Applications, Springer, vol. 8(1), pages 1-19, December.
    4. Farouk Mselmi, 2022. "Generalized linear model for subordinated Lévy processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 772-801, June.
    5. 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.
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