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Multivariate insurance models: an overview. (English) Zbl 1284.91197
Summary: This literature review summarizes the results from a collection of research papers that relate to modeling insurance claims and the processes associated with them. We consider work by more than 55 authors, published or presented between 1971 and 2008.

MSC:
91B30 Risk theory, insurance (MSC2010)
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
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[1] Ambagaspitiya, R.S., Compound bivariate Lagrangian Poisson distributions, Insurance: mathematics and economics, 23, 21-31, (1998) · Zbl 0984.62038
[2] Ambagaspitiya, R.S., On the distribution of a sum of correlated aggregate claims, Insurance: mathematics and economics, 23, 15-19, (1998) · Zbl 0916.62072
[3] Ambagaspitiya, R.S., On the distributions of two classes of correlated aggregate claims, Insurance: mathematics and economics, 24, 301-308, (1999) · Zbl 0945.62110
[4] Bauerle, N.; Grubel, R., Multivariate counting processes: copulas and beyond, Astin bulletin, 25, 379-408, (2005) · Zbl 1098.62132
[5] Bauerle, N.; Grubel, R., Multivariate risk processes with interacting intensities, Advanced applied probability, 40, 578-601, (2008) · Zbl 1152.60058
[6] Bauerle, N.; Muller, A., Modeling and comparing dependencies in multivariate risk portfolios, Astin bulletin, 28, 59-76, (1998) · Zbl 1137.91484
[7] Cai, J.; Li, H., Multivariate risk model of phase type, Insurance: mathematics and economics, 36, 2, 137-152, (2005) · Zbl 1122.60049
[8] Cossette, H.; Gaillardetz, P.; Marceau; Rioux, J., On two dependent individual risk models, Insurance: mathematics and economics, 30, 153-166, (2002) · Zbl 1055.91044
[9] Cossette, H.; Marceau, E., The discrete-time risk model with correlated classes of business, Insurance: mathematics and economics, 26, 133-149, (2000) · Zbl 1103.91358
[10] Dhaene, J.; Denuit, M., The safest dependence structure among risks, Insurance: mathematics and economics, 25, 1, 11-21, (1999) · Zbl 1072.62651
[11] Dhaene, J.; Goovaerts, M., On the dependency of risks in the individual life risk model, Insurance: mathematics and economics, 19, 243-253, (1997) · Zbl 0931.62089
[12] Enachescu, D.; Vernic, R., Approximating the distribution of a randomly weighted sum of random variables using kernel and neural network methods, Mathematical reports, 8, 58, 37-47, (2006) · Zbl 1150.62446
[13] Frees, E.W.; Valdez, E.A., Hierarchical insurance claims modeling, Journal of the American statistical association, 103, 484, 1457-1469, (2008) · Zbl 1286.62087
[14] Frostig, E., A comparison between homogeneous and heterogeneous portfolios, Insurance: mathematics and economics, 29, 1, 59-71, (2001) · Zbl 1072.91026
[15] Frostig, E., Comparison of portfolios which depend on multivariate Bernoulli random variables with fixed marginals, Insurance: mathematics and economics, 29, 3, 319-332, (2001), Papers presented at the 4th IME Conference, Universitat de Barcelona, Barcelona, 24-26 July 2001 · Zbl 1074.91551
[16] Genest, C.; Marceau; Mesfioui, M., Compound Poisson approximations for individual models with dependent risks, Insurance: mathematics and economics, 32, 73-91, (2003) · Zbl 1055.91050
[17] Goff, M., Multivariate discrete phase-type distributions with their applications, (2001), Academic Press
[18] Gregory, G.G., On bivariate cumulative damage models with application, Biometrical journal, 44, 835-849, (2002)
[19] Hawkes, A.G., Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58, 83-90, (1971) · Zbl 0219.60029
[20] Hu, T.; Wu, Z., On dependence of risks and stop-loss premiums, Insurance: mathematics and economics, 24, 3, 323-332, (1999) · Zbl 0945.62109
[21] Ivanova, N.; Khokhlov, Y., Reconstruction multivariate distribution with Poisson marginals, Vestnik of Moscow university, 15, 33-37, (2001) · Zbl 0996.60004
[22] Ivanova, N.; Khokhlov, Y., Multidimensional collective risk model, Vestnik of Moscow university, 15, 35-43, (2003), (published 2005, 2003 version referenced)
[23] Kolev, M.; Paiva, D., Random sums of exchangeable variables and actuarial applications, Insurance: mathematics and economics, 42, 147-153, (2008) · Zbl 1141.91521
[24] Lindskog, F.; McNeil, A.J., Common Poisson shock models: applications to insurance and credit risk modelling, Astin bulletin, 33, 209-238, (2003) · Zbl 1087.91030
[25] Marshall, A.W.; Olkin, I., Inequalities: theory of majorizations and its applications, (1979), Academic Press
[26] Meng, Q.; Zhang, X.; Guo, J., On a risk model with dependence between claim sizes and claim intervals, Statistics and probability letters, 78, 1727-1734, (2008) · Zbl 1284.91258
[27] Muller, A., Stop-loss order for portfolios of dependent risks, Insurance: mathematics and economics, 21, 3, 219-223, (1997) · Zbl 0894.90022
[28] Ozaki, T., Maximum likelihood estimation of hawkes’ self-exciting point processes, Annual institute of statistical mathematics, 31, 145-155, (1979) · Zbl 0447.62081
[29] Panjer, H.H., Recursive evaluation of a family of compound distributions, Astin bulletin (international acturial association), 12, 22-26, (1981)
[30] Pfeifer, D.; Neslehova, J., Modeling and generating dependent risk processes for irm and dfa, Astin bulletin, 34, 333-360, (2004) · Zbl 1159.91410
[31] Ribas, C.; Marin-Solano, J.; Alegre, A., On the computation of the aggregate claims distribution in the individual life model with bivariate dependencies, Insurance: mathematics and economics, 32, 201-215, (2003) · Zbl 1025.62042
[32] Rodrigues, J.; Cid, J.E.R.; Achcar, J.A., Bayesian analysis for the superposition of two dependent non-homogeneous Poisson processes, Communications in statistics, 31, 1467-1478, (2002) · Zbl 1008.62030
[33] Vernic, R., On the bivariate generalized Poisson distribution, Astin bulletin, 27, 23-31, (1997)
[34] Vernic, R., Recursive evaluation of some bivariate compound distributions, Astin bulletin, 29, 315-325, (1999)
[35] Vernic, R., A multivariate generalization of the generalized Poisson distribution, Astin bulletin, 30, 57-67, (2000) · Zbl 1114.62332
[36] Vernic, R., Arithmetizing continuous distributions: numerical aspects, Universitatea “ovidius” constanta, 11, 189-194, (2003) · Zbl 1084.62508
[37] Vernic, R., On a bivariate generalisation of sundt’s class of counting distributions, Math reports, 56, 185-201, (2004) · Zbl 1137.62309
[38] Wang, G.; Yuen, K.C., On a correlated aggregate claims model with thinning-dependence structure, Insurance: mathematics and economics, 36, 456-468, (2005) · Zbl 1120.62095
[39] Yan, L.; Wenquan, Y.; Yijun, H., On the ruin function for a correlated aggregate claims model with Poisson and Erlang risk processes, Acta Mathematica scientia, 26, 321-330, (2006) · Zbl 1137.91498
[40] Yuen, K.C.; Guo, J.; Wu, X., On a correlated aggregate claims model with Poisson and Erlang risk processes, Insurance: mathematics and economics, 31, 205-214, (2002) · Zbl 1074.91566
[41] Yuen, K.C., Wang, G., 2001. Comparing two models with dependent classes of business, in: Proceedings of the 36th Actuarial Research Conference.
[42] Yuen, K.C.; Wu, X., A discrete-time risk model with interaction between classes of business, Insurance: mathematics and economics, 33, 117-133, (2003) · Zbl 1074.91031
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