Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T16:16:25.400Z Has data issue: false hasContentIssue false
  • English
  • Français

EFFICIENT ESTIMATION OF ERLANG MIXTURES USING iSCAD PENALTY WITH INSURANCE APPLICATION

Published online by Cambridge University Press:  13 May 2016

Cuihong Yin  and
X. Sheldon Lin
Cuihong Yin
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, China E-Mail: [email protected]
X. Sheldon Lin*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, China Department of Statistical Sciences, University of Toronto, Toronto, Ontario, CanadaM5S 3G3
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Erlang mixture model has been widely used in modeling insurance losses due to its desirable distributional properties. In this paper, we consider the problem of efficient estimation of the Erlang mixture model. We present a new thresholding penalty function and a corresponding EM algorithm to estimate model parameters and to determine the order of the mixture. Using simulation studies and a real data application, we demonstrate the efficiency of the EM algorithm.

Keywords

Erlang mixtureEM algorithmiSCAD penaltyVaRTVaR
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 46 , Issue 3 , September 2016 , pp. 779 - 799
Copyright
Copyright © Astin Bulletin 2016 

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

Badescu, A.L., Gong, L., Lin, X.S. and Tang, D. (2015) Modeling correlated frequencies with application in operational risk management. Journal of Operational Risk, 10 (1), 1–43.CrossRefGoogle Scholar
Barges, M., Loisel, S. and Venel, X. (2013) On finite-time ruin probabilities with reinsurance cycles influenced by large claims. Scandinavian Actuarial Journal, 2013 (3), 163–185.CrossRefGoogle Scholar
Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2006) Statistics of Extremes: Theory and Applications. John Wiley & Sons: Chichester, England.Google Scholar
Cebrian, A. C., Denuit, M. and Lambert, P. (2003) Generalized Pareto fit to the society of actuaries large claims database. North American Actuarial Journal, 7 (3), 18–36.CrossRefGoogle Scholar
Chen, J. and Khalili, A. (2008) Order selection in finite mixture models with a nonsmooth penalty. Journal of the American Statistical Association, 103 (484), 1674–1683.CrossRefGoogle Scholar
Chen, J. and Li, P. (2009) Hypothesis test for normal mixture models: The EM approach. Annals of Statistics, 37 (5A), 2523–2542.CrossRefGoogle Scholar
Chen, J., Li, P. and Fu, Y. (2012) Inference on the order of a normal mixture. Journal of the American Statistical Association, 107 (499), 1096–1105.CrossRefGoogle Scholar
Cossette, H., Mailhot, M. and Marceau, E. (2012) TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts. Insurance: Mathematics and Economics, 50 (2), 247–256.Google Scholar
Cossette, H., Cote, M. P., Marceau, E. and Moutanabbir, K. (2013) Multivariate distribution defined with Farlie-Gumbel-Morgenstern copula and mixed Erlang marginals: aggregation and capital allocation. Insurance: Mathematics and Economics, 52 (3), 560–572.Google Scholar
Donoho, D. L. and Johnstone, I. M. (1994) Threshold selection for wavelet shrinkage of noisy data. In Engineering Advances: New Opportunities for Biomedical Engineers, Proceedings of the 16th Annual International Conference of the IEEE, A24–A25. Baltimore, MD.Google Scholar
Fan, J. and Li, R. (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American statistical Association, 96 (456), 1348–1360.CrossRefGoogle Scholar
Frank, L. E. and Friedman, J. H. (1993) A statistical view of some chemometrics regression tools. Technometrics, 35 (2), 109–135.CrossRefGoogle Scholar
Gong, L., Badescu, A. L. and Cheung, E. C. (2012) Recursive methods for a multi-dimensional risk process with common shocks. Insurance: Mathematics and Economics, 50 (1), 109–120.Google Scholar
Hashorva, E. and Ratovomirija, G. (2015) On Sarmanov mixed Erlang risks in insurance applications. ASTIN Bulletin, 45 (01), 175–205.CrossRefGoogle Scholar
Kass, R. E. and Wasserman, L. (1995) A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. Journal of the American Statistical Association, 90 (431), 928–934.CrossRefGoogle Scholar
Keribin, C. (2000) Consistent estimation of the order of mixture models. The Indian Journal of Statistics, Series A 62 (1), 49–66.Google Scholar
Landriault, D. and Willmot, G.E. (2009) On the joint distributions of the time to ruin, the surplus prior to ruin, and the deficit at ruin in the classical risk model. North American Actuarial Journal, 13 (2), 252–270.CrossRefGoogle Scholar
Lee, S.C. and Lin, X.S. (2010) Modeling and evaluating insurance losses via mixtures of Erlang distributions. North American Actuarial Journal, 14 (1), 107–130.CrossRefGoogle Scholar
Lee, S.C. and Lin, X.S. (2012) Modeling dependent risks with multivariate Erlang mixtures. ASTIN Bulletin, 42 (01), 153–180.Google Scholar
Lin, X.S. and Willmot, G.E. (2000) The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 27 (1), 19–44.Google Scholar
Porth, L., Zhu, W. and Tan, K.S. (2014) A credibility-based Erlang mixture model for pricing crop reinsurance. Agricultural Finance Review, 74 (2), 162–187.CrossRefGoogle Scholar
Teicher, H. (1963) Identifiability of finite mixtures. Annals of Mathematical Statistics, 34 (4), 1265–1269.CrossRefGoogle Scholar
Tibshirani, R. (1996) Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B (Methodological), 58 (1), 267–288.Google Scholar
Tijms, H.C. (2003) A First Course in Stochastic Models. John Wiley and Sons.CrossRefGoogle Scholar
Tsai, C.C.L. and Willmot, G. E. (2002) On the moments of the surplus process perturbed by diffusion. Insurance: Mathematics and Economics, 31 (3), 327–350.Google Scholar
Verbelen, R., Antonio, K., Badescu, A., Gong, L. and Lin, X.S. 2015. Fitting mixtures of Erlangs to censored and truncated data using the EM algorithm. ASTIN Bulletin, 45 (3), 729–758.CrossRefGoogle Scholar
Verbelen, R., Antonio, K. and Claeskens, G. (2016) Multivariate mixtures of Erlangs for density estimation under censoring. Lifetime Data Analysis, to appear.CrossRefGoogle Scholar
Willmot, G.E. and Woo, J.K. (2015) On some properties of a class of multivariate Erlang mixtures with insurance applications. ASTIN Bulletin, 45 (1), 151–173.CrossRefGoogle Scholar
Yakowitz, S.J. and Spragins, J.D. (1968) On the identifiability of finite mixtures. The Annals of Mathematical Statistics, 39 (1), 209–214.CrossRefGoogle Scholar
Yin, C. and Lin, X.S. (2016) On the consistency of the order of Erlang mixtures, working paper.Google Scholar