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Modelling censored losses using splicing: a global fit strategy with mixed Erlang and extreme value distributions. (English) Zbl 1404.62115

Summary: In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME) distribution for the body and a Pareto distribution for the tail. This combines the flexibility of the ME distribution with the ability of the Pareto distribution to model extreme values. We extend our splicing approach for censored and/or truncated data. Relevant examples of such data can be found in financial risk analysis. We illustrate the flexibility of this splicing model using practical examples from risk measurement.

MSC:

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

Software:

R; evir; Interval; ADGofTest
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References:

[1] Aban, I. B.; Meerschaert, M. M.; Panorska, A. K., Parameter estimation for the truncated Pareto distribution, J. Amer. Statist. Assoc., 101, 473, 270-277, (2006) · Zbl 1118.62312
[2] Akaike, H., A new look at the statistical model identification, IEEE Trans. Automat. Control, 19, 6, 716-723, (1974) · Zbl 0314.62039
[3] Albrecher, H.; Beirlant, J.; Teugels, J., Reinsurance: Actuarial and Statistical Aspects, (2017), John Wiley & Sons, Ltd Chichester, UK · Zbl 1376.91004
[4] Antonio, K.; Plat, R., Micro-level stochastic loss reserving for general insurance, Scand. Actuar. J., 2014, 7, 649-669, (2014) · Zbl 1401.91091
[5] Aue, F.; Kalkbrener, M., LDA at work: deutsche bank’s approach to quantifying operational risk, J. Oper. Risk, 1, 4, 49-93, (2006)
[6] Babu, G. J.; Rao, C. R., Goodness-of-fit tests when parameters are estimated, Sankhyā, 66, 1, 63-74, (2004) · Zbl 1192.62126
[7] Bakar, S. A.A.; Hamzah, N. A.; Maghsoudi, M.; Nadarajah, S., Modeling loss data using composite models, Insurance Math. Econom., 61, 1146-1154, (2015) · Zbl 1314.91130
[8] Beirlant, J.; Fraga Alves, M. I.; Gomes, M. I., Tail Fitting for truncated and non-truncated Pareto-type distributions, Extremes, 19, 3, 429-462, (2016) · Zbl 1360.62244
[9] Beirlant, J.; Goegebeur, Y.; Teugels, J.; Segers, J., (Statistics of Extremes: Theory and Applications, Wiley Series in Probability and Statistics, (2004), John Wiley & Sons, Ltd. Chichester, UK)
[10] Beirlant, J.; Guillou, A.; Dierckx, G.; Fils-Villetard, A., Estimation of the extreme value index and extreme quantiles under random censoring, Extremes, 10, 3, 151-174, (2007) · Zbl 1157.62027
[11] Brazauskas, V.; Kleefeld, A., Modeling severity and measuring tail risk of Norwegian fire claims, N. Am. Actuar. J., 20, 1, 1-16, (2016)
[12] Calderín-Ojeda, E.; Kwok, C. F., Modeling claims data with composite stoppa models, Scand. Actuar. J., 2016, 9, 817-836, (2016) · Zbl 1401.62205
[13] Cao, R.; Vilar, J. M.; Devía, A., Modelling consumer credit risk via survival analysis, SORT, 33, 1, 3-30, (2009) · Zbl 1274.91454
[14] Ciumara, R., An actuarial model based on the composite Weibull-Pareto distribution, Math. Rep. (Bucur.), 8, 4, 401-414, (2006) · Zbl 1120.62332
[15] Cooray, K.; Ananda, M. M., Modeling actuarial data with a composite lognormal-Pareto model, Scand. Actuar. J., 2005, 5, 321-334, (2005) · Zbl 1143.91027
[16] Dempster, A. P.; Laird, N. M.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc. Ser. B. Stat. Methodol., 39, 1, 1-38, (1977) · Zbl 0364.62022
[17] Einmahl, J. H.; Fils-Villetard, A.; Guillou, A., Statistics of extremes under random censoring, Bernoulli, 14, 1, 207-227, (2008) · Zbl 1155.62036
[18] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling Extremal Events for Insurance and Finance, (1997), Springer-Verlag Berlin Heidelberg
[19] Fackler, M., 2013. Reinventing Pareto: Fits For Both Small and Large Losses. ASTIN Colloquium, Den Haag.
[20] Fay, M. P.; Shaw, P. A., Exact and asymptotic weighted logrank tests for interval censored data: the interval R package, J. Stat. Softw., 36, 2, 1-34, (2010)
[21] Gil Bellosta, C.J., 2011. ADGofTest: Anderson-Darling GoF test. R package version 0.3.
[22] Hill, B. M., A simple general approach to inference about the tail of a distribution, Ann. Statist., 3, 5, 1163-1174, (1975) · Zbl 0323.62033
[23] Kaplan, E. L.; Meier, P., Nonparametric estimation from incomplete observations, J. Amer. Statist. Assoc., 53, 282, 457-481, (1958) · Zbl 0089.14801
[24] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., (Loss Models: From Data to Decisions, Wiley Series in Probability and Statistics, (2012), John Wiley & Sons, Inc. Hoboken, NJ) · Zbl 1272.62002
[25] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., (Loss Models: Further Topics, Wiley Series in Probability and Statistics, (2013), John Wiley & Sons, Inc. Hoboken, NJ) · Zbl 1273.62008
[26] Lee, D.; Li, W. K.; Wong, T. S.T., Modeling insurance claims via a mixture exponential model combined with peaks-over-threshold approach, Insurance Math. Econom., 51, 3, 538-550, (2012) · Zbl 1285.91061
[27] Lee, S. C.K.; Lin, X. S., Modeling and evaluating insurance losses via mixtures of Erlang distributions, N. Am. Actuar. J., 14, 1, 107-130, (2010)
[28] Massart, P., The tight constant in the dvoretzky-kiefer-wolfowitz inequality, Ann. Probab., 18, 3, 1269-1283, (1990) · Zbl 0713.62021
[29] McNeil, A. J., Estimating the tails of loss severity distributions using extreme value theory, Astin Bull., 27, 1, 117-137, (1997)
[30] McNeil, A. J.; Frey, R.; Embrechts, P., (Quantitative Risk Management: Concepts, Techniques and Tools, Princeton Series in Finance, (2005), Princeton University Press Princeton, NJ) · Zbl 1089.91037
[31] Miljkovic, T.; Grün, B., Modeling loss data using mixtures of distributions, Insurance Math. Econom., 70, 387-396, (2016) · Zbl 1373.62527
[32] Nadarajah, S.; Bakar, S. A.A., New composite models for the danish fire insurance data, Scand. Actuar. J., 2014, 2, 180-187, (2014) · Zbl 1401.91177
[33] Neuts, M. F., Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, (1981), John Hopkins University Press Baltimore, MD · Zbl 0469.60002
[34] Panjer, H. H., (Operational Risk: Modeling Analytics, Wiley Series in Probability and Statistics, (2006), John Wiley & Sons, Inc. Hoboken, NJ) · Zbl 1258.62101
[35] Peters, G. W.; Shevchenko, P. V., Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk, (2015), John Wiley & Sons, Inc. Hoboken, NJ
[36] Pfaff, B., McNeil, A., 2012. evir: Extreme Values in . package version 1.7-3.
[37] Pigeon, M.; Denuit, M., Composite lognormal-Pareto model with random threshold, Scand. Actuar. J., 2011, 3, 177-192, (2011) · Zbl 1277.62258
[38] R: A Language and Environment for Statistical Computing, (2017), R Foundation for Statistical Computing Vienna, Austria
[39] Rytgaard, M., 1996. Simulation experiments on the mean residual lifetime function. In: Proceedings of the XXVII ASTIN Colloquium, Copenhagen, Denmark,pp. 59-81.
[40] Schwarz, G. E., Estimating the dimension of a model, J. Amer. Statist. Assoc., 6, 2, 461-464, (1978) · Zbl 0379.62005
[41] Scollnik, D. P.M., On composite lognormal-Pareto models, Scand. Actuar. J., 2007, 1, 20-33, (2007) · Zbl 1146.91028
[42] Scollnik, D. P.M.; Sun, C., Modeling with Weibull-Pareto models, N. Am. Actuar. J., 16, 2, 260-272, (2012) · Zbl 1291.62186
[43] Teodorescu, S.; Panaitescu, E., On the truncated composite Weibull-Pareto model, Math. Rep. (Bucur.), 11, 61, 259-273, (2009) · Zbl 1199.60041
[44] Tijms, H. C., (Stochastic Models: an Algorithmic Approach, Wiley Series in Probability and Statistics, (1994), John Wiley & Sons, Ltd. Chichester, UK) · Zbl 0838.60075
[45] Turnbull, B. W., The empirical distribution function with arbitrarily grouped, censored and truncated data, J. R. Stat. Soc. Ser. B. Stat. Methodol., 38, 3, 290-295, (1976) · Zbl 0343.62033
[46] Verbelen, R.; Antonio, K.; Claeskens, G., Multivariate mixtures of erlangs for density estimation under censoring, Lifetime Data Anal., 22, 3, 429-455, (2016) · Zbl 1422.62194
[47] Verbelen, R.; Gong, L.; Antonio, K.; Badescu, A.; Lin, S., Fitting mixtures of erlangs to censored and truncated data using the EM algorithm, Astin Bull., 45, 3, 729-758, (2015) · Zbl 1390.62227
[48] Willmot, G. E.; Lin, X. S., Risk modelling with the mixed Erlang distribution, Appl. Stoch. Models Bus. Ind., 27, 1, 2-16, (2011)
[49] Willmot, G. E.; Woo, J., On the class of Erlang mixtures with risk theoretic applications, N. Am. Actuar. J., 11, 2, 99-115, (2007)
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