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Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications

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  • Gómez-Déniz, Emilio
  • Sarabia, José Mari­a
  • Calderi­n-Ojeda, Enrique

Abstract

In this paper we propose a new compound negative binomial distribution by mixing the p negative binomial parameter with an inverse Gaussian distribution and where we consider the reparameterization p=exp(-[lambda]). This new formulation provides a tractable model with attractive properties which make it suitable for application not only in the insurance setting but also in other fields where overdispersion is observed. Basic properties of the new distribution are studied. A recurrence for the probabilities of the new distribution and an integral equation for the probability density function of the compound version, when the claim severities are absolutely continuous, are derived. A multivariate version of the new distribution is proposed. For this multivariate version, we provide marginal distributions, the means vector, the covariance matrix and a simple formula for computing multivariate probabilities. Estimation methods are discussed. Finally, examples of application for both univariate and bivariate cases are given.

Suggested Citation

  • Gómez-Déniz, Emilio & Sarabia, José Mari­a & Calderi­n-Ojeda, Enrique, 2008. "Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 39-49, February.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:39-49
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    References listed on IDEAS

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    1. Lemaire, Jean, 1979. "How to Define a Bonus-Malus System with an Exponential Utility Function," ASTIN Bulletin, Cambridge University Press, vol. 10(3), pages 274-282, December.
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    Cited by:

    1. Tzougas, George & Yik, Woo Hee & Mustaqeem, Muhammad Waqar, 2019. "Insurance ratemaking using the Exponential-Lognormal regression model," LSE Research Online Documents on Economics 101729, London School of Economics and Political Science, LSE Library.
    2. Tzougas, George & Hoon, W. L. & Lim, J. M., 2019. "The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking," LSE Research Online Documents on Economics 101728, London School of Economics and Political Science, LSE Library.
    3. George Tzougas, 2020. "EM Estimation for the Poisson-Inverse Gamma Regression Model with Varying Dispersion: An Application to Insurance Ratemaking," Risks, MDPI, vol. 8(3), pages 1-23, September.
    4. Spark C. Tseung & Ian Weng Chan & Tsz Chai Fung & Andrei L. Badescu & X. Sheldon Lin, 2022. "A Posteriori Risk Classification and Ratemaking with Random Effects in the Mixture-of-Experts Model," Papers 2209.15212, arXiv.org.
    5. Tzougas, George, 2020. "EM estimation for the Poisson-Inverse Gamma regression model with varying dispersion: an application to insurance ratemaking," LSE Research Online Documents on Economics 106539, London School of Economics and Political Science, LSE Library.
    6. Tzougas, George & Pignatelli di Cerchiara, Alice, 2021. "The multivariate mixed Negative Binomial regression model with an application to insurance a posteriori ratemaking," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 602-625.
    7. Gómez-Déniz, Emilio & Sarabia, José María & Calderín-Ojeda, Enrique, 2011. "A new discrete distribution with actuarial applications," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 406-412, May.

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