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Remarks on a generalized inverse Gaussian type integral with applications

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  • Willmot, Gordon E.
  • Woo, Jae-Kyung

Abstract

In this paper, we consider the truncated Inverse Gaussian (IG) distribution and the generalized Inverse Gaussian (GIG) distribution and then obtain the components in its generalized Esscher transform and size-biased Esscher transform. Consequently, this enables us to derive an explicit expression for the cumulative distribution function of the GIG distribution with a half-integer parameter. We show that this result has applications for the evaluation of the mixed Poisson with the truncated GIG-type distribution, Tail Value-at-Risk for GIG risk, and for a Sparre Andersen risk model.

Suggested Citation

  • Willmot, Gordon E. & Woo, Jae-Kyung, 2022. "Remarks on a generalized inverse Gaussian type integral with applications," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    • Handle: RePEc:eee:apmaco:v:430:y:2022:i:c:s0096300322003769
    DOI: 10.1016/j.amc.2022.127302
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    References listed on IDEAS

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    1. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    2. Chen, Ying & Härdle, Wolfgang & Jeong, Seok-Oh, 2008. "Nonparametric Risk Management With Generalized Hyperbolic Distributions," Journal of the American Statistical Association, American Statistical Association, vol. 103(483), pages 910-923.
    3. Rigby, R.A. & Stasinopoulos, D.M. & Akantziliotou, C., 2008. "A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 381-393, December.
    4. Gordon Willmot & Jae-Kyung Woo, 2007. "On the Class of Erlang Mixtures with Risk Theoretic Applications," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 99-115.
    5. Landsman, Zinoviy & Valdez, Emiliano A., 2005. "Tail Conditional Expectations for Exponential Dispersion Models," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 189-209, May.
    6. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    7. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 459-465, February.
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    Cited by:

    1. Yuanchuang Shan & Huisheng Shu & Haoran Yi, 2023. "Pricing Equity-Indexed Annuities under a Stochastic Dividend Model," Mathematics, MDPI, vol. 11(3), pages 1-12, January.

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