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Fitting mixtures of Erlangs to censored and truncated data using the EM algorithm. (English) Zbl 1390.62227

Summary: 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.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62N01 Censored data models
91B30 Risk theory, insurance (MSC2010)

Software:

ElemStatLearn; plfit
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References:

[1] Akaike, H., A new look at the statistical model identification, IEEE Transactions on Automatic Control, 19, 716-723, (1974) · Zbl 0314.62039
[2] Antonio, K.; Plat, R., Micro-level stochastic loss reserving for general insurance, Scandinavian Actuarial Journal, 2014, 649-669, (2014) · Zbl 1401.91091
[3] Badescu, A.; Gong, L.; Lin, X. S.; Tang, D., Modeling correlated frequencies with applications in operational risk management, Journal of Operational Risk, 10, 1-43, (2015)
[4] Beirlant, J.; Goegebeur, Y.; Segers, J.; Teugels, J.; De Waal, D.; Ferro, C., Statistics of Extremes: Theory and Applications, (2004), Chichester, UK: Wiley, Chichester, UK · Zbl 1070.62036
[5] Bolancé, C.; Guillén, M.; Gustafsson, J.; Nielsen, J. P., Quantitative Operational Risk Models, (2012), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 1233.91003
[6] Cameron, A.; Trivedi, P., Microeconometrics: Methods and Applications, (2005), New York: Cambridge University Press, New York · Zbl 1156.62092
[7] Chernobai, A.; Rachev, S.; Fabozzi, F.; Lee, C.-F.; Lee, J. C., Handbook of Financial Econometrics and Statistics, Composite goodness-of-fit tests for left-truncated loss samples, 575-596, (2014), New York: Springer, New York
[8] Clauset, A.; Shalizi, C. R.; Newman, M. E., Power-law distributions in empirical data, SIAM Review, 51, 661-703, (2009) · Zbl 1176.62001
[9] Dempster, A. P.; Laird, N. M.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society. Series B (Methodological), 39, 1-38, (1977) · Zbl 0364.62022
[10] Dufour, R.; Maag, U., Distribution results for modified Kolmogorov-Smirnov statistics for truncated or censored, Technometrics, 20, 29-32, (1978) · Zbl 0397.62026
[11] Frees, E. W.; Valdez, E. A., Hierarchical insurance claims modeling, Journal of the American Statistical Association, 103, 1457-1469, (2008) · Zbl 1286.62087
[12] Guilbaud, O., Exact Kolmogorov-type tests for left-truncated and/or right-censored data, Journal of the American Statistical Association, 83, 213-221, (1988) · Zbl 0666.62047
[13] Hastie, T. J.; Tibshirani, R. J.; Friedman, J., The Elements of Statistical Learning: Data Mining, Inference, and Prediction, (2009), Heidelberg: Springer-Verlags, Heidelberg · Zbl 1273.62005
[14] Kaplan, E. L.; Meier, P., Nonparametric estimation from incomplete observations, Journal of the American statistical association, 53, 457-481, (1958) · Zbl 0089.14801
[15] Klugman, S.; Rioux, J., Toward a unified approach to fitting loss models, North American Actuarial Journal, 10, 63-83, (2006)
[16] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models: From Data to Decisions, (2012), Hoboken, NJ: John Wiley & Sons, Inc, Hoboken, NJ · Zbl 1272.62002
[17] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models: Further Topics, (2013), Hoboken, NJ: John Wiley & Sons, Inc, Hoboken, NJ · Zbl 1273.62008
[18] Lee, D.; Li, W. K.; Wong, T. S.T., Modeling insurance claims via a mixture exponential model combined with peaks-over-threshold approach, Insurance: Mathematics and Economics, 51, 538-550, (2012) · Zbl 1285.91061
[19] Lee, G.; Scott, C., EM algorithms for multivariate Gaussian mixture models with truncated and censored data, Computational Statistics & Data Analysis, 56, 2816-2829, (2012) · Zbl 1255.62308
[20] Lee, S. C.; Lin, X. S., Modeling and evaluating insurance losses via mixtures of Erlang distributions, North American Actuarial Journal, 14, 107-130, (2010)
[21] Lee, S. C.; Lin, X. S., Modeling dependent risks with multivariate Erlang mixtures, ASTIN Bulletin, 42, 153-180, (2012) · Zbl 1277.62255
[22] Mccall, B. P., Unemployment insurance rules, joblessness, and part-time work, Econometrica, 64, 647, (1996) · Zbl 0847.90039
[23] Mclachlan, G. J.; Jones, P., Fitting mixture models to grouped and truncated data via the em algorithm, Biometrics, 44, 571-578, (1988) · Zbl 0707.62214
[24] Mclachlan, G. J.; Peel, D., Finite mixture models, (2001), Hoboken, NJ: John Wiley & Sons, Inc, Hoboken, NJ
[25] Mclachlan, G. J.; Krishnan, T., The EM Algorithm and Extensions, (2008), Hoboken, NJ: John Wiley & Sons, Inc, Hoboken, NJ · Zbl 1165.62019
[26] Neuts, M. F., Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach, (1981), Baltimore, MD: The John Hopkins University Press, Baltimore, MD · Zbl 0469.60002
[27] Pigeon, M.; Denuit, M., Composite lognormal-Pareto model with random threshold, Scandinavian Actuarial Journal, 2011, 177-192, (2011) · Zbl 1277.62258
[28] Schwarz, G., Estimating the dimension of a model, The Annals of Statistics, 6, 461-464, (1978) · Zbl 0379.62005
[29] Tijms, H. C., Stochastic Models: An Algorithmic Approach, (1994), Hoboken, NJ: John Wiley & Sons, Inc, Hoboken, NJ · Zbl 0838.60075
[30] Willmot, G. E.; Lin, X. S., Risk modelling with the mixed Erlang distribution, Applied Stochastic Models in Business and Industry, 27, 2-16, (2011)
[31] Willmot, G. E.; Woo, J.-K., On the class of Erlang mixtures with risk theoretic applications, North American Actuarial Journal, 11, 99-115, (2007)
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