Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T03:05:48.818Z Has data issue: false hasContentIssue false
  • English
  • Français

A Bayesian Approach for Estimating Extreme Quantiles Under a Semiparametric Mixture Model

Published online by Cambridge University Press:  09 August 2013

Stefano Cabras  and
María Eugenia Castellanos
María Eugenia Castellanos
Affiliation:
Department of Statistics and Operations Research, Rey Juan Carlos University (Spain). C/ Tulipán, 28933, Móstoles (Spain)., E-Mail: maria.castellanos@urjc.es
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we propose an additive mixture model, where one component is the Generalized Pareto distribution (GPD) that allows us to estimate extreme quantiles. GPD plays an important role in modeling extreme quantiles for the wide class of distributions belonging to the maximum domain of attraction of an extreme value model. One of the main difficulty with this modeling approach is the choice of the threshold u, such that all observations greater than u enter into the likelihood function of the GPD model. Difficulties are due to the fact that GPD parameter estimators are sensible to the choice of u. In this work we estimate u, and other parameters, using suitable priors in a Bayesian approach. In particular, we propose to model all data, extremes and non-extremes, using a semiparametric model for data below u, and the GPD for the exceedances over u. In contrast to the usual estimation techniques for u, in this setup we account for uncertainty on all GPD parameters, including u, via their posterior distributions. A Monte Carlo study shows that posterior credible intervals also have frequentist coverages. We further illustrate the advantages of our approach on two applications from insurance.

Keywords

Extreme valuesGeneralized Pareto distributionJeffreys' priorLindsey methodMixture distributionSemiparametric density estimation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 1 , May 2011 , pp. 87 - 106
Copyright
Copyright © International Actuarial Association 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Behrens, C.N., Lopes, H.F. and Gamerman, D. (2004) Bayesian analysis of extreme events with threshold estimation. Statistical Modelling 4, 227–244.CrossRefGoogle Scholar
Beirlant, J., Teugels, J. and Vynckier, P. (1996) Practical Analysis of Extreme Values. Leuven University Press, Leuven, Belgium.Google Scholar
Brazauskas, V. and Serfling, R. (2003) Favorable estimators for fitting pareto models. A study using goodness-of-fit measures with actual data. ASTIN Bulletin 33(2), 365–381.CrossRefGoogle Scholar
Cabras, S. and Morales, J. (2007) Extreme value analysis within a parametric outlier detection framework. Applied Stochastic Models in Business and Industry 23(2), 157–164.CrossRefGoogle Scholar
Castellanos, M. and Cabras, S. (2007) A default bayesian procedure for the generalized pareto distribution. Journal of Statistical Planning and Inference 137(2), 473–483.CrossRefGoogle Scholar
Castillo, E. and Hadi, A. (1997) Fitting the generalized pareto distribution to data. Journal of the American Statistical Association 92, 1609–1620.CrossRefGoogle Scholar
Chavez-Demoulin, and Embrechts, P.V. (2009) An EVT primer for credit risk. Oxford University Press.Google Scholar
Choulakian, V. and Stephens, M. (2001) Goodness-of-fit tests for the generalized pareto distribution. Technometrics 43(4), 478–484.CrossRefGoogle Scholar
Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. New York: Springer-Verlag.CrossRefGoogle Scholar
Davison, A. (1984) Statistical extremes and Applications, Chapter Modelling excesses over high thresholds, with an application, pp. 461–482. Reidel, Dordrecht.CrossRefGoogle Scholar
de Zea Bermudez, P. and Turkman, M.A. (2003) Bayesian approach to parameter estimation of the generalized pareto distribution. Test 12, 259–277.CrossRefGoogle Scholar
de Zea Bermudez, P., Turkman, M.A. and Turkman, K. (2001) A predictive approach to tail probability estimation. Extremes 4, 295–314.CrossRefGoogle Scholar
Donnelly, C. and Embrechts, P. (2010) The devil is in the tails: actuarial mathematics and the subprime mortgage crisis. ASTIN Bulletin 40(1), 1–33.CrossRefGoogle Scholar
Efron, B. and Tibshirani, R. (1996) Using specially designed exponential families for density estimation. The Annals of Statistics 24(6), 2431–2461.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modeling extremal events for insurance and finance. New York: Springer.CrossRefGoogle Scholar
Frigessi, A., Haug, O. and Rue, H. (2002) A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes 5, 219–235.CrossRefGoogle Scholar
Grimshaw, S.D. (1993) Computing maximum likelihood estimates for the generalized pareto distribution. Technometrics 35(2), 185–191.CrossRefGoogle Scholar
Hosking, J. and Wallis, J. (1987) Parameter and quantile estimation for the generalized pareto distribution. Technometrics 29(3), 339–349.CrossRefGoogle Scholar
Lindsey, J. (1974a) Comparison of probability distributions. Journal of the Royal Statistical Society, Series B 36, 38–47.Google Scholar
Lindsey, J. (1974b) Construction and comparison of statistical models. Journal of the Royal Statistical Society, Series B 36, 418–425.Google Scholar
McNeil, A.J. (1997) Estimating the tails of loss severity distributions using extreme value theory. ASTIN Bulletin, 117–137.Google Scholar
Mikosch, T. (2003) Modeling dependence and tails of financial time series. In Finkenstaedt, B. and Rootzen, H. (Eds.), Extreme Values in Finance, Telecommunications, and the Environment, pp. 185–286. Chapman & Hall/CRC.Google Scholar
Neves, C. and Fraga Alves, M.I. (2004) Reiss and thomas' automatic selection of the number of extremes. Computational Statistics and Data Analysis 47, 689–704.CrossRefGoogle Scholar
Padgett, W. and Johnson, M. (1983) Some bayesian lower bounds on reliability in the lognormal distribution. The Canadian Journal of Statistics 11, 137–147.CrossRefGoogle Scholar
Pickands, J. (1975) Statistical inference using extreme order statistics. The Annals of Statistics 3, 119–131.Google Scholar
R Development Core Team (2009) R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0.Google Scholar
Reiss, R.D. and Thomas, M. (2007) Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields (Third ed.). Birkhäuser Verlag.Google Scholar
Roeder, K. and Wasserman, L. (1997) Practical bayesian density estimation using mixtures of normals. Journal of American Statistical Association 92, 894–902.CrossRefGoogle Scholar
Smith, R. (2003) Extreme values in finance, telecommunications and the environment, Chapter 1. Statistics of extremes, with applications in environment, insurance and finance. Chapman and Hall/CRC Press.Google Scholar
Smith, R.L. (1984) Statistical extremes and Applications, Chapter Threshold methods for sample extremes, pp. 621–638. Reidel, Dordrecht.CrossRefGoogle Scholar
Sun, D. and Berger, J. (1994) Bayesian sequential reliability for weibull and related distributions. Ann. Inst. Statist. Math. 46(2), 221–249.CrossRefGoogle Scholar
Tancredi, A., Anderson, C. and O'Hagan, A. (2006) Accounting for threshold uncertainty in extreme value estimation. Extremes 9, 87–106.CrossRefGoogle Scholar
Ventura, L., Cabras, S. and Racugno, W. (2009) Prior distributions from pseudo-likelihoods in the presence of nuisance parameters. Journal of the American Statistical Association 104(486), 768–774.CrossRefGoogle Scholar