Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-18T02:42:25.363Z Has data issue: false hasContentIssue false
  • English
  • Français

FITTING MIXTURES OF ERLANGS TO CENSORED AND TRUNCATED DATA USING THE EM ALGORITHM

Published online by Cambridge University Press:  20 July 2015

Roel Verbelen ,
Lan Gong ,
Katrien Antonio ,
Andrei Badescu  and
Sheldon Lin
Roel Verbelen*
Affiliation:
LStat, Faculty of Economics and Business, KU Leuven, Belgium
Lan Gong
Affiliation:
Department of Statistical Sciences, University of Toronto, Canada E-Mail: lan.gong@tdsecurities.com
Katrien Antonio
Affiliation:
LStat, Faculty of Economics and Business, KU Leuven, Belgium Faculty of Economics and Business, University of Amsterdam, the Netherlands E-Mail: Katrien.Antonio@kuleuven.be
Andrei Badescu
Affiliation:
Department of Statistical Sciences, University of Toronto, Canada E-Mail: badescu@utstat.toronto.edu
Sheldon Lin
Affiliation:
Department of Statistical Sciences, University of Toronto, Canada E-Mail: sheldon@utstat.toronto.edu
*
E-Mail: roel.verbelen@kuleuven.be
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss how to fit mixtures of Erlangs to censored and truncated data by iteratively using the EM algorithm. Mixtures of Erlangs form a very versatile, yet analytically tractable, class of distributions making them suitable for loss modeling purposes. The effectiveness of the proposed algorithm is demonstrated on simulated data as well as real data sets.

Keywords

Mixture of Erlang distributions with a common scale parametercensoringtruncationexpectation–maximization algorithmmaximum likelihood
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 3 , September 2015 , pp. 729 - 758
Copyright
Copyright © Astin Bulletin 2015 

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

Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19 (6), 716–723.CrossRefGoogle Scholar
Antonio, K. and Plat, R. (2014) Micro-level stochastic loss reserving for general insurance. Scandinavian Actuarial Journal, 2014 (7), 649–669.CrossRefGoogle Scholar
Badescu, A., Gong, L., Lin, X.S. and Tang, D. (2015) Modeling correlated frequencies with applications in operational risk management. Journal of Operational Risk, 10 (1), 1–43.CrossRefGoogle Scholar
Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J., De Waal, D. and Ferro, C. (2004) Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. Chichester, UK: Wiley.CrossRefGoogle Scholar
Bolancé, C., Guillén, M., Gustafsson, J. and Nielsen, J.P. (2012) Quantitative Operational Risk Models. Boca Raton, FL: CRC Press.CrossRefGoogle Scholar
Cameron, A. and Trivedi, P. (2005) Microeconometrics: Methods and Applications. New York: Cambridge University Press.CrossRefGoogle Scholar
Chernobai, A., Rachev, S. and Fabozzi, F. (2014) Composite goodness-of-fit tests for left-truncated loss samples. In Handbook of Financial Econometrics and Statistics (eds. Lee, C.-F. and Lee, J.C.), pp. 575–596. New York: Springer.Google Scholar
Clauset, A., Shalizi, C.R. and Newman, M.E. (2009) Power-law distributions in empirical data. SIAM Review, 51 (4), 661–703.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 (Methodological), 39 (1), 1–38.CrossRefGoogle Scholar
Dufour, R. and Maag, U. (1978) Distribution results for modified Kolmogorov–Smirnov statistics for truncated or censored. Technometrics, 20 (1), 29–32.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
Guilbaud, O. (1988) Exact Kolmogorov-type tests for left-truncated and/or right-censored data. Journal of the American Statistical Association, 83 (401), 213–221.CrossRefGoogle Scholar
Hastie, T.J., Tibshirani, R.J. and Friedman, J. (2009) The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed.Heidelberg: Springer-Verlags.CrossRefGoogle Scholar
Kaplan, E.L. and Meier, P. (1958) Nonparametric estimation from incomplete observations. Journal of the American statistical association, 53 (282), 457–481.CrossRefGoogle Scholar
Klugman, S. and Rioux, J. (2006) Toward a unified approach to fitting loss models. North American Actuarial Journal, 10 (1), 63–83.CrossRefGoogle Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2012) Loss Models: From Data to Decisions, volume 715. Hoboken, NJ: John Wiley & Sons, Inc.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2013) Loss Models: Further Topics. Hoboken, NJ: John Wiley & Sons, Inc.CrossRefGoogle Scholar
Lee, D., Li, W.K. and Wong, T.S.T. (2012) Modeling insurance claims via a mixture exponential model combined with peaks-over-threshold approach. Insurance: Mathematics and Economics, 51 (3), 538–550.Google Scholar
Lee, G. and Scott, C. (2012) EM algorithms for multivariate Gaussian mixture models with truncated and censored data. Computational Statistics & Data Analysis, 56 (9), 2816–2829.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 (1), 153–180.Google Scholar
McCall, B.P. (1996) Unemployment insurance rules, joblessness, and part-time work. Econometrica, 64 (3), 647–82.CrossRefGoogle Scholar
McLachlan, G.J. and Jones, P. (1988) Fitting mixture models to grouped and truncated data via the em algorithm. Biometrics, 44 (2), 571–578.CrossRefGoogle ScholarPubMed
McLachlan, G.J. and Peel, D. (2001) Finite mixture models. Hoboken, NJ: John Wiley & Sons, Inc.Google Scholar
McLachlan, G.J. and Krishnan, T. (2008) The EM Algorithm and Extensions, volume 382. Hoboken, NJ: John Wiley & Sons, Inc.CrossRefGoogle Scholar
Neuts, M.F. (1981) Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach. Baltimore, MD: The John Hopkins University Press.Google Scholar
Pigeon, M. and Denuit, M. (2011) Composite lognormal-Pareto model with random threshold. Scandinavian Actuarial Journal, 2011 (3), 177–192.CrossRefGoogle Scholar
Schwarz, G. (1978) Estimating the dimension of a model. The Annals of Statistics, 6 (2), 461–464.CrossRefGoogle Scholar
Tijms, H.C. (1994) Stochastic Models: An Algorithmic Approach. Hoboken, NJ: John Wiley & Sons, Inc.Google Scholar
Willmot, G.E. and Lin, X.S. (2011) Risk modelling with the mixed Erlang distribution. Applied Stochastic Models in Business and Industry, 27 (1), 2–16.CrossRefGoogle Scholar
Willmot, G.E. and Woo, J.-K. (2007) On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal, 11 (2), 99–115.CrossRefGoogle Scholar