×

zbMATH — the first resource for mathematics

Extending composite loss models using a general framework of advanced computational tools. (English) Zbl 1422.91351
Summary: Composite models have a long history in actuarial science because they provide a flexible method of curve-fitting for heavy-tailed insurance losses. The ongoing research in this area continuously suggests methodological improvements for existing composite models and considers new composite models. A number of different composite models have been previously proposed in the literature to fit the popular data set related to Danish fire losses. This paper provides the most comprehensive analysis of composite loss models on the Danish fire losses data set to date by evaluating 256 composite models derived from 16 parametric distributions that are commonly used in actuarial science. If not suitably addressed, inevitable computational challenges are encountered when estimating these composite models that may lead to sub-optimal solutions. General implementation strategies are developed for parameter estimation in order to arrive at an automatic way to reach a viable solution, regardless of the specific head and/or tail distributions specified. The results lead to an identification of new well-fitting composite models and provide valuable insights into the selection of certain composite models for which the tail-evaluation measures can be useful in making risk management decisions.
Reviewer: Reviewer (Berlin)

MSC:
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62G32 Statistics of extreme values; tail inference
91-08 Computational methods for problems pertaining to game theory, economics, and finance
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akaike, H., A new look at the statistical model identification, IEEE Transactions on Automatic Control, 19, 6, 716-723, (1974) · Zbl 0314.62039
[2] Bakar, S. A. A.; Hamzaha, N. A.; Maghsoudia, M.; Nadarajah, S., Modeling loss data using composite models, Insurance: Mathematics and Economics, 61, 146-154, (2015) · Zbl 1314.91130
[3] Blostein, M.; Miljkovic, T., On modeling left-truncated loss data using mixtures of distributions, Insurance: Mathematics and Economics, 85, 35-46, (2019) · Zbl 1415.62076
[4] 2017pracma: practical numerical math functions. R package version 2.1.1.
[5] Brazauskas, V.; Kleefeld, A., Modeling severity and measuring tail risk of Norwegian fire claims, North American Actuarial Journal, 20, 1, 1-16, (2016) · Zbl 1414.62415
[6] Burnham, K. P.; Anderson, D. R., Model selection and multimodel inference: a practical information-theoretic approach, (2003), New York: Springer Verlag, New York
[7] Calderín-Ojeda, E.; Kwok, C. F., Modeling claims data with composite Stoppa models, Scandinavian Actuarial Journal, 2016, 817-836, (2016) · Zbl 1401.62205
[8] Chen, Y.; Miljkovic, T., From grouped to de-grouped data: a new approach in distribution fitting for grouped data, Journal of Statistical Computation and Simulation, 89, 2, 272-291, (2019)
[9] Ciumara, R., An actuarial model based on the composite Weibull-Pareto distribution, Mathematical Reports-Bucharest, 8, 4, 401-414, (2006) · Zbl 1120.62332
[10] Cooray, K., The Weibull-Pareto composite family with applications to the analysis of unimodal failure rate data, Communications in Statistics - Theory and Methods, 38, 11, 1901-1915, (2009) · Zbl 1167.62021
[11] Cooray, K.; Ananda, M. M., Modeling actuarial data with a composite lognormal-Pareto model, Scandinavian Actuarial Journal, 2005, 5, 321-334, (2005) · Zbl 1143.91027
[12] 2013SMPracticals: practicals for use with Davison (2003) ‘Statistical Models’. R package version 1.4-2.
[13] Embrechts, P.; Puccetti, G.; Rüschendorf, L.; Wang, R.; Beleraj, A., An academic response to Basel 3.5, Risks, 2, 1, 25-48, (2014)
[14] 2016NumDeriv: accurate numerical derivatives. R package version 2016.8-1.
[15] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss models: from data to decisions, (2012), Hobuken, NJ: John Wiley & Sons, Hobuken, NJ · Zbl 1272.62002
[16] Lee, S. C. K.; Lin, X. S., Modeling and evaluating insurance losses via mixtures of Erlang distributions, North American Actuarial Journal, 14, 1, 107-130, (2010)
[17] Miljkovic, T.; Barabanov, N., Modeling veterans – health benefit grants using the expectation maximization algorithm, Journal of Applied Statistics, 42, 6, 1166-1182, (2015)
[18] Miljkovic, T.; Grün, B., Modeling loss data using mixtures of distributions, Insurance: Mathematics and Economics, 70, 387-396, (2016) · Zbl 1373.62527
[19] Miljkovic, T.; Orr, M., An evaluation of the reconstructed coefficient of determination and potential adjustments, Communications in Statistics - Simulation and Computation, 46, 9, 6705-6718, (2017) · Zbl 06849544
[20] Nadarajah, S.; Bakar, S., New composite models for the Danish fire insurance data, Scandinavian Actuarial Journal, 2014, 180-187, (2014) · Zbl 1401.91177
[21] Pigeon, M.; Denuit, M., Composite lognormal-Pareto model with random threshold, Scandinavian Actuarial Journal, 2011, 177-192, (2011) · Zbl 1277.62258
[22] Punzo, A.; Bagnato, L.; Maruotti, A., Compound unimodal distributions for insurance losses, Insurance: Mathematics and Economics, 81, 95-107, (2018) · Zbl 1416.91217
[23] 2018R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.
[24] Sarabia, J. M.; Calderín-Ojeda, E., Analytical expressions of risk quantities for composite models, Journal of Risk Model Validation, 12, 3, 75-101, (2018)
[25] Schwarz, G., Estimating the dimension of a model, The Annals of Statistics, 6, 2, 461-464, (1978) · Zbl 0379.62005
[26] Scollnik, D., On composite lognormal-Pareto models, Scandinavian Actuarial Journal, 2007, 20-33, (2007) · Zbl 1146.91028
[27] Scollnik, D. P.; Sun, C., Modeling with Weibull-Pareto models, North American Actuarial Journal, 16, 260-272, (2012) · Zbl 1291.62186
[28] Sherlock, C.; Fearnhead, P.; Roberts, G. O., The random walk Metropolis: linking theory and practice through a case study, Statistical Science, 25, 2, 172-190, (2010) · Zbl 1328.60177
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.