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The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking

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A Discussion on recent papers to this article was published on 01 June 2019

Abstract

This paper presents the Negative Binomial-Inverse Gaussian regression model for approximating the number of claims as an alternative to mixed Poisson regression models that have been widely used in various disciplines including actuarial applications. The Negative Binomial-Inverse Gaussian regression model can be considered as a plausible model for highly dispersed claim count data and this is the first time that it is used in a statistical or actuarial context. The main achievement is that we propose a quite simple Expectation-Maximization type algorithm for maximum likelihood estimation of the model. Finally, a real data application using motor insurance data is examined and both the a priori and a posteriori, or Bonus-Malus, premium rates resulting from the Negative Binomial-Inverse Gaussian model are calculated via the net premium principle and compared to those determined by the Negative Binomial Type I and the Poisson-Inverse Gaussian regression models that have been traditionally used for a priori and a posteriori ratemaking.

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Notes

  1. Note that the NBIG distribution given in Eq. (5) is different from the one used in Gómez-Deniz et al. [24], who considered the case without covariate information.

  2. Note that pseudo-values will be used in lieu of the original values to estimate the parameters of interest.

  3. Note that the location and scale parameters of the NBI and PIG models are denoted by \(\mu\) and \(\sigma\) respectively

  4. Note that all the explanatory variables and the parameters of the models are statistically significant at a 5% threshold.

  5. We also used three fourths of the data set to estimate the parameters of the models and the remaining one fourth was used to test the out-of-sample prediction accuracy of the models. As expected, our findings were consistent with those provided by the AIC criterion. For more details, refer to Stone [46] who showed that AIC and leave-one out cross validation are asymptotically equivalent.

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Acknowledgements

The research reported here has been supported by LSE LIFE and LSE Teaching and Learning Centre. We would like to thank the two anonymous referees for their constructive comments and suggestions that have greatly improved the paper. We also would like to thank the participants at the 10th International Conference of the ERCIM WG on Computational and Methodological Statistics. The usual disclaimer applies.

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Authors and Affiliations

  1. Department of Statistics, London School of Economics and Political Science, London, UK

    G. Tzougas, W. L. Hoon & J. M. Lim

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  1. G. Tzougas
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  2. W. L. Hoon
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  3. J. M. Lim
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Correspondence to G. Tzougas.

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Tzougas, G., Hoon, W.L. & Lim, J.M. The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking. Eur. Actuar. J. 9, 323–344 (2019). https://doi.org/10.1007/s13385-018-0186-2

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