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Skew mixture models for loss distributions: A Bayesian approach

Author

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  • Bernardi, Mauro
  • Maruotti, Antonello
  • Petrella, Lea

Abstract

The derivation of loss distribution from insurance data is a very interesting research topic but at the same time not an easy task. To find an analytic solution to the loss distribution may be misleading although this approach is frequently adopted in the actuarial literature. Moreover, it is well recognized that the loss distribution is strongly skewed with heavy tails and presents small, medium and large size claims which hardly can be fitted by a single analytic and parametric distribution. Here we propose a finite mixture of Skew Normal distributions that provides a better characterization of insurance data. We adopt a Bayesian approach to estimate the model, providing the likelihood and the priors for the all unknown parameters; we implement an adaptive Markov Chain Monte Carlo algorithm to approximate the posterior distribution. We apply our approach to a well known Danish fire loss data and relevant risk measures, such as Value-at-Risk and Expected Shortfall probability, are evaluated as well.

Suggested Citation

  • Bernardi, Mauro & Maruotti, Antonello & Petrella, Lea, 2012. "Skew mixture models for loss distributions: A Bayesian approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 617-623.
  • Handle: RePEc:eee:insuma:v:51:y:2012:i:3:p:617-623
    DOI: 10.1016/j.insmatheco.2012.08.002
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    References listed on IDEAS

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    1. Bernardi, Mauro, 2013. "Risk measures for skew normal mixtures," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1819-1824.
    2. Ahn, Soohan & Kim, Joseph H.T. & Ramaswami, Vaidyanathan, 2012. "A new class of models for heavy tailed distributions in finance and insurance risk," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 43-52.
    3. Burnecki, Krzysztof & Misiorek, Adam & Weron, Rafal, 2010. "Loss Distributions," MPRA Paper 22163, University Library of Munich, Germany.
    4. Francesco Lagona & Marco Picone, 2012. "Model-based clustering of multivariate skew data with circular components and missing values," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(5), pages 927-945, September.
    5. Eling, Martin, 2012. "Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 239-248.
    6. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika van der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639.
    7. McNeil, Alexander J., 1997. "Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 27(01), pages 117-137, May.
    8. repec:dau:papers:123456789/6069 is not listed on IDEAS
    9. Bolance, Catalina & Guillen, Montserrat & Pelican, Elena & Vernic, Raluca, 2008. "Skewed bivariate models and nonparametric estimation for the CTE risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 386-393, December.
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    Citations

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    Cited by:

    1. Bernardi, Mauro, 2013. "Risk measures for skew normal mixtures," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1819-1824.
    2. Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
    3. Valeria Bignozzi & Claudio Macci & Lea Petrella, 2017. "Large deviations for risk measures in finite mixture models," Papers 1710.03252, arXiv.org, revised Feb 2018.
    4. Peng, Zuoxiang & Li, Chunqiao & Nadarajah, Saralees, 2016. "Extremal properties of the skew-t distribution," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 10-19.
    5. Farias, Rafael B.A. & Montoril, Michel H. & Andrade, José A.A., 2016. "Bayesian inference for extreme quantiles of heavy tailed distributions," Statistics & Probability Letters, Elsevier, vol. 113(C), pages 103-107.
    6. Loperfido, Nicola, 2014. "A note on the fourth cumulant of a finite mixture distribution," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 386-394.
    7. repec:eee:insuma:v:79:y:2018:i:c:p:247-259 is not listed on IDEAS
    8. Mauro Bernardi & Ghislaine Gayraud & Lea Petrella, 2013. "Bayesian inference for CoVaR," Papers 1306.2834, arXiv.org, revised Nov 2013.
    9. Abu Bakar, S.A. & Hamzah, N.A. & Maghsoudi, M. & Nadarajah, S., 2015. "Modeling loss data using composite models," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 146-154.
    10. repec:eee:empfin:v:43:y:2017:i:c:p:1-32 is not listed on IDEAS

    More about this item

    Keywords

    Markov chain Monte Carlo; Bayesian analysis; Mixture model; Skew-Normal distributions; Loss distribution; Danish data;

    JEL classification:

    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics

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