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AN EM ALGORITHM FOR FITTING A NEW CLASS OF MIXED EXPONENTIAL REGRESSION MODELS WITH VARYING DISPERSION

Published online by Cambridge University Press:  08 May 2020

George Tzougas Open the ORCID record for George Tzougas [Opens in a new window]  and
Dimitris Karlis
George Tzougas*
Affiliation:
Department of Statistics, London School of Economics and Political Science, LondonWC2A 2AE, UK, E-mail: [email protected]
Dimitris Karlis
Affiliation:
Department of Statistics, Athens University of Economics and Business, Greece, E-mail: [email protected]
*
E-mail: [email protected]
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Abstract

Regression modelling involving heavy-tailed response distributions, which have heavier tails than the exponential distribution, has become increasingly popular in many insurance settings including non-life insurance. Mixed Exponential models can be considered as a natural choice for the distribution of heavy-tailed claim sizes since their tails are not exponentially bounded. This paper is concerned with introducing a general family of mixed Exponential regression models with varying dispersion which can efficiently capture the tail behaviour of losses. Our main achievement is that we present an Expectation-Maximization (EM)-type algorithm which can facilitate maximum likelihood (ML) estimation for our class of mixed Exponential models which allows for regression specifications for both the mean and dispersion parameters. Finally, a real data application based on motor insurance data is given to illustrate the versatility of the proposed EM-type algorithm.

Keywords

Mixed exponential distributionsEM algorithmregression models for the mean and dispersionparametersnon-life insuranceheavy-tailed losses
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 2 , May 2020 , pp. 555 - 583
Copyright
© Astin Bulletin 2020

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References

Ahn, S., Kim, J.H. and Ramaswami, V. (2012) A new class of models for heavy tailed distributions in finance and insurance risk. Insurance: Mathematics and Economics, 51(1), 43–52.Google Scholar
Barreto-Souza, W. and Simas, A.B. (2016) General mixed Poisson regression models with varying dispersion. Statistics and Computing, 26(6), 1263–1280.CrossRefGoogle Scholar
Beirlant, J., Derveaux, V., De Meyer, A.M., Goovaerts, M.J., Labie, E. and Maenhoudt, B. (1992) Statistical risk evaluation applied to (Belgian) car insurance. Insurance: Mathematics and Economics, 10(4), 289–302.Google Scholar
Bhattacharya, S.K. and Kumar, S. (1986) E-IG model in life testing. Calcutta Statistical Association Bulletin, 35(1–2), 85–90.CrossRefGoogle Scholar
Bladt, M. and Rojas-Nandayapa, L. (2018) Fitting phase–type scale mixtures to heavy–tailed data and distributions. Extremes, 21(2), 285–313.CrossRefGoogle Scholar
Booth, J.G. and Hobert, J.P. (1999) Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(1), 265–285.CrossRefGoogle Scholar
Booth, J.G., Hobert, J.P. and Jank, W. (2001) A survey of Monte Carlo algorithms for maximizing the likelihood of a two-stage hierarchical model. Statistical Modelling, 1(4), 333–349.CrossRefGoogle Scholar
Burnham, K.P. and Anderson, D.R. (2002) Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach. New York: Springer.Google Scholar
Calderín-Ojeda, E., Fergusson, K. and Wu, X. (2017) An EM algorithm for double-Pareto-lognormal generalized linear model applied to heavy-tailed insurance claims. Risks, 5(4), 60.CrossRefGoogle Scholar
Cole, T.J. and Green, P.J. (1992) Smoothing reference centile curves: The LMS method and penalized likelihood. Statistics in Medicine, 11(10), 1305–1319.CrossRefGoogle ScholarPubMed
De Jong, P. and Heller, G.Z. (2008) Generalized Linear Models for Insurance Data, Vol. 10. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 39(1), 1–38.Google Scholar
Denuit, M., Marechal, X., Pitrebois, S. and Walhin, J.F. (2007) Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems. Chichester, West Sussex: John Wiley and Sons.CrossRefGoogle Scholar
Dunn, P.K. and Smyth, G.K. (1996) Randomized quantile residuals. Computational and Graphical Statistics, 5(3), 236–245.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events. Applications of Mathematics, Vol. 33. Berlin, Heidelberg: Springer-Verlag.Google Scholar
Frangos, N. and Vrontos, S. (2001) Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin, 31(1) 1–22.CrossRefGoogle Scholar
Frangos, N. and Karlis, D. (2004) Modelling losses using an exponential-inverse Gaussian distribution. Insurance: Mathematics and Economics, 35(1), 53–67.Google Scholar
Frees, E.W. and Valdez, E.A. (2008) Hierarchical insurance claims modeling Journal of the American Statistical Association, 103(484), 1457–1469.CrossRefGoogle Scholar
Frees, E.W., Derrig, R.A. and Meyers, G. (2014a) Predictive Modeling Applications in Actuarial Science, Vol. 1. New York: Cambridge University Press.CrossRefGoogle Scholar
Frees, E.W., Derrig, R.A. and Meyers, G. (2014b) Predictive Modeling Applications in Actuarial Science, Vol. 2. New York: Cambridge University Press.CrossRefGoogle Scholar
Heller, G.Z., Stasinopoulos, M.D., Rigby, R.A. and De Jong, P. (2007) Mean and dispersion modeling for policy claims costs. Scandinavian Actuarial Journal, 4, 281–292.CrossRefGoogle Scholar
Hesselager, O., Wang, S. and Willmot, G. (1998) Exponential and scale mixtures and equilibrium distributions. Scandinavian Actuarial Journal, 2, 125–142.CrossRefGoogle Scholar
Hürlimann, W. (2014) Pareto type distributions and excess-of-loss reinsurance. International Journal of Research and Reviews in Applied Sciences, 18(3), 1.Google Scholar
Jones, B.L. and Zitikis, R. (2003) Empirical estimation of risk measures and related quantities. North American Actuarial Journal, 7(4), 44–54.CrossRefGoogle Scholar
Jørgensen, B. (1982) Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics, Vol. 9.New York: Springer-Verlag.Google Scholar
Karlis, D. (2001) A general EM approach for maximum likelihood estimation in mixed Poisson regression models. Statistical Modelling, 1(4), 305–318.CrossRefGoogle Scholar
Karlis, D. (2005) EM algorithm for mixed Poisson and other discrete distributions. ASTIN Bulletin, 35(1), 3–24.CrossRefGoogle Scholar
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Hoboken, New Jersey: John Wiley and Sons.CrossRefGoogle Scholar
Klein, N., Denuit, M., Lang, S. and Kneib, T. (2014) Nonlife ratemaking and risk management with Bayesian generalized additive models for location, scale, and shape. Insurance: Mathematics and Economics, 55, 225–249.Google Scholar
Klugman, S., Panjer, H. and Willmot, G. (2012) Loss Models: From Data to Decisions, Vol. 715. New York: John Wiley and Sons.Google Scholar
Louis, T.A. (1982) Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 44, 226–233.Google Scholar
McLachlan, G. and Krishnan, T. (2007) The EM Algorithm and Extensions, Vol. 382. Hoboken, New Jersey: John Wiley and Sons.Google Scholar
Mencía, J. and Sentana, E. (2005) Estimation and testing of dynamic models with generalized hyperbolic innovations. CEPR Discussion Papers, No. 5177.Google Scholar
Raftery, A.E. (1995) Bayesian model selection in social research. Sociological Methodology, 25, 111–163.CrossRefGoogle Scholar
Ramirez-Cobo, P., Lillo, R.E., Wilson, S. and Wiper, M.P. (2010) Bayesian inference for double Pareto lognormal queues. The Annals of Applied Statistics, 4(3), 1533–1557.CrossRefGoogle Scholar
Rigby, R.A. and Stasinopoulos, D.M. (1996a) A semi-parametric additive model for variance heterogeneity. Statistics and Computing, 6(1), 57–65.CrossRefGoogle Scholar
Rigby, R.A. and Stasinopoulos, D.M. (1996b) Mean and dispersion additive models. In Statistical Theory and Computational Aspects of Smoothing (eds. Härdle, W. and Schimek, M.G.), pp. 215–230. Heidelberg: Physica.CrossRefGoogle Scholar
Rigby, R.A. and Stasinopoulos, D.M. (2005) Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(3), 507–554.CrossRefGoogle Scholar
Stasinopoulos, D.M., Rigby, B. and Akantziliotou, C. (2008) Instructions on How to Use the Gamlss Package in R, 2nd edn. http://www.gamlss.org.Google Scholar
Rosenberg, M.A., Frees, E.W., Sun, J., Johnson, P.H. and Robinson, J.M. (2007) Predictive modeling with longitudinal data: A case study with Wisconsin nursing homes North American Actuarial Journal, 11(3), 54–69.CrossRefGoogle Scholar
Seshadri, V. (2012) The Inverse Gaussian Distribution: Statistical Theory and Applications, Vol. 137. New York: Springer Science and Business Media.Google Scholar
Stasinopoulos, M.D., Rigby, R.A., Heller, G.Z., Voudouris, V. and De Bastiani, F. (2017) Flexible Regression and Smoothing: Using GAMLSS in R. Florida: Chapman and Hall/CRC.CrossRefGoogle Scholar
Tzougas, G., Vrontos, S. and Frangos, N. (2014) Optimal bonus-malus systems using finite mixture models. ASTIN Bulletin, 44(2), 417–444.CrossRefGoogle Scholar
Tzougas, G., Vrontos, S. and Frangos, N. (2018) Bonus-malus systems with two component mixture models arising from different parametric families. North American Actuarial Journal, 22(1), 55–91.CrossRefGoogle Scholar
Tzougas, G., Yik, W.H. and Mustaqeem, M.W. (2019) Insurance ratemaking using the exponential-lognormal regression model. Annals of Actuarial Science, 1–30. doi: 10.1017/S1748499519000034.Google Scholar
Wills, M., Valdez, E. and Frees, E. (2006) GB2 regression with insurance claim severities. Conference Paper, UNSW Actuarial Research Symposium 2008, Sydney, New South Wales, Australia.Google Scholar
Wang, S. (1998) An actuarial index of the right-tail risk. North American Actuarial Journal, 2(2), 88–101.CrossRefGoogle Scholar
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