Bernardi, Mauro and Maruotti, Antonello and Lea, Petrella (2012): Skew mixture models for loss distributions: a Bayesian approach.
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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 mislading 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 present 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 unknow 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, as ValueatRisk and Expected Shortfall probability, are evaluated as well.
Item Type:  MPRA Paper 

Original Title:  Skew mixture models for loss distributions: a Bayesian approach 
Language:  English 
Keywords:  Markov chain Monte Carlo, Bayesian analysis, mixture model, SkewNormal distributions, Loss distribution, Danish data 
Subjects:  C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C52  Model Evaluation, Validation, and Selection C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C11  Bayesian Analysis: General C  Mathematical and Quantitative Methods > C0  General > C01  Econometrics 
Item ID:  39826 
Depositing User:  Mauro Bernardi 
Date Deposited:  04. Jul 2012 12:29 
Last Modified:  07. Sep 2015 06:24 
References:  Ahn, S., Kim J. H. T., and Ramaswami, V., (2012). A new class of models for heavy tailed distributions in finance and insurance risk. \emph{Insurance: Mathematics and Economics}, 51, 4352. Andrieu, C., and Moulines \'{E}., (2006). On the ergodicity properties of some adaptive MCMC algorithms. \emph{Annals of Applied Probability}, 16, 14621505. Andrieu C. and Thoms, J., (2008). A tutorial on adaptive MCMC. \emph{Statistical and Computing}, 18, 343373. Azzalini, A., (1985). A class of distributions which includes the Normal ones. \emph{Scandinavian Journal of Statistics}, 12, 171178. Bayes, C. L. and Branco, M. D., (2007). Bayesian inference for the skewness parameter of the scalar Skew Normal distribution. \emph{Brazilian Journal of Probability and Statistics}, 21, 141163. Bernardi M., (2012). Risk measures for SkewNormal mixtures. Working paper, MEMOTEF, Sapienza University of Rome. Bernardi M., and Petrella, L., (2012). Parallel adaptive MCMC with applications. Proceedings of the 46th Italian Statistical Society Meeting. Bernardo, J. M., (2005). Reference Analysis. In \emph{Handbook of Statistics, 25}, Elsevier, NorthHolland, Amsterdam, 459507. Bolance, C., Guillen, M., Pelican, E., and Vernic, R., (2008). Skewed bivariate models and nonparametric estimation for the CTE risk measure. \emph{Insurance: Mathematics and Economics}, 43, 386393. Burnecki, K., Misiorek, A., and Weron, R., (2010). Loss distributions. In \emph{Statistical Tools for Finance and Insurance}. Cizek, P., H\"{a}rdle, W. K., and Weron, R. Eds. SpringerVerlag. Celeux, G., Hurn, M. N., and Robert, C. P., (2000). Computational and inferential difficulties with mixture posterior distributions. \emph{Journal of the American Statistical Association}, 95, 957979. Chipman, H., George E. I., and McCulloch E., (2001). The practical implementation of Bayesian model selection. \textit{IMS Lecture Notes  Monograph series (2001)}, 38. Cooray, K., and Amanda, M. A., (2005). Modeling actuarial data with a composite Lognormal Pareto model. \emph{Scandinavian Actuarial Journal}, 5, 321334. Diebold, J. and Robert, C. P., (1994). Estimation of finite mixture distributions through Bayesian sampling. \emph{Biometrika}, 56, 363375. Eling, M., (2012). Fitting insurance claims to skewed distributions: Are the SkewNormal and SkewStudent good models? \emph{Insurance: Mathematics and Economics}, 51, 239248. Embrechts, P., Kl\"{u}ppelberg, C., and Mikosch, T., (1997). \emph{Modelling Extremal Events for Insurance and Finance}. SpringerVerlag, New York. Frigessi, A. and Haug O. and Rue, H., (2002). A dynamic mixture model for unsupervised tail estimation without threshold selection. \emph{Extremes}, 5, 219235. Fr\"{u}hwirthSchnatter, S., (2006). \emph{Finite Mixture and Markov Switching Models}. Springer Series in Statistics. Springer, New York. Fr\"{u}hwirthSchnatter, S., and Pyne, S., (2010). Bayesian inference for finite mixtures of univariate and multivariate SkewNormal and Skew$t$ distributions. \emph{Biostatistics}, 11, 31736. Genton, M. G., (2004). \emph{SkewElliptical distriutions and their applications: a journey beyond normality}. Chapman and Hall. Geweke, J., (1992). Evaluating the Accuracy of SamplingBased Approaches to the Calculation of Posterior Moments. In J. M. Bernardo, J. Berger, A. P. Dawid, and A. F. M. Smith, eds., \emph{Bayesian Statistics 4}, Oxford University Press, pp. 169193. Geweke, J., (2005). \emph{Contemporary Bayesian Econometrics and Statistics}. Wiley Series in Probability and Statistics. Wiley, Hoboken. Haario H., Saksman E. and Tamminen J., (2001). An Adaptive Metropolis algorithm. \emph{Bernoulli}, 14, 223242. Hastings W. K., (1970). Monte Carlo sampling methods using Markov chains and their applications. \emph{Biometrika}, 57, 97109. Jasra, A., Holmes, C. C., and Stephens, D. A., (2005), Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modelling, \emph{Statistical Science}, 20, 5067. Kass, R. E., and Raftery A. E., (1995). Bayes factors. \textit{Journal of the American Statistical Association}, 90, 773795. Liseo, B., and Loperfido, M. A., (2006). A note on reference priors for the scalar Skew Normal distribution. \textit{Journal of Statistical Planning and Inference}, 136, 373389. Lagona, F., and Picone, M., (2012). Modelbased clustering of multivariate skew data with circular components and missing values, \emph{Journal of Applied Statistics}, 39, 927945. Marin, J. M., Mengersen, K., Robert, C. P., (2005). Bayesian modelling and Inference on mixtures of distributions. In \emph{Handbook of Statistics, 25}, Elsevier, NorthHolland, Amsterdam, 459507. McNeil, A. J., (1997). Estimating the Tails of Loss Severity Distributions using Extreme Value Theory, \emph{Astin Bulletin}, 27, 117137. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). Equations of state calculations by fast computing machines. \emph{Journal of Chemical Physics}, 21, 10871091. Neal R. M., (2001). Annealed importance sampling. \emph{Statistics and Computing}, 11, 125139. Richardson, S., and Green P. J., (1997). On Bayesian analysis of mixtures with an unknown number of components. {\em Journal of the Royal Statistical Society Series B}, 59, 731758. Robert, C. P., (1996), Mixtures of distributions: Inference and estimation. In \emph{Markov Chain Monte Carlo in Practice}. Gilks, W. R., Richardson S., and Spiegelhalter D. J. Eds. Chapman and Hall. Robert, C. P., and Casella, G., (2004). \emph{Monte Carlo Statistical Methods}. Springer Texts in Statistics. Springer, New York. Robbins H. and Monro S., (1951). A stochastic approximation method. \emph{Annals of Mathematical Stastistics}, 22, 400407. Sahu, S. K., Dey, D. K., and Branco, M. D., (2003). A new class of multivariate skew distributions with applications to Bayesian regression models. \emph{Canadian Journal of Statistics}, 31, 129150. Sattayatham P. and Talangtam T., (2012). Fitting of finite mixture distributions to motor insurance claims. \emph{Journal of Mathematics and Stastics}, 8, 4956. Scollnik, D. P. M., (2007). On composite lognormalPareto models. \emph{Scandinavian Actuarial Journal}, 1, 2033. Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. \emph{Journal of the Royal Stastical Society, Series B}, 59, 731792. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/39826 
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