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Predictive compound risk models with dependence. (English) Zbl 1454.91195

In the collective risk model for a given period of time, the sequence \(\{C_{j}\}_{j\in{\mathbb N}}\) of individual losses is i.i.d.and independent of the number of claims \(N\) and the total loss is defined as \(S=\sum_{j=1}^{N}C_{j}\). In the present paper, the authors consider a related and rather particular multiperiod model in which independence between individual losses and the number of claims is replaced by an assumption on the structure of \(E(S|N)\). For the total loss in a future period they propose a predictor which depends on claims experience in the past.

MSC:

91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics

Citations:

Zbl 1373.62515
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References:

[1] Antonio, K.; Valdez, E. A., Statistical concepts of a priori and a posteriori risk classification in insurance, Adv. Statist. Anal., 96, 2, 187-224 (2012) · Zbl 1443.62328
[2] Boucher, J.-P.; Denuit, M.; Guillén, M., Models of insurance claim counts with time dependence based on generalization of Poisson and negative binomial distributions, Variance, 2, 1, 135-162 (2008)
[3] Dionne, G.; Vanasse, C., A generalization of automobile insurance rating models: the negative binomial distribution with a regression component, ASTIN Bull.: J. IAA, 19, 2, 199-212 (1989)
[4] Doss, D., Definition and characterization of multivariate negative binomial distribution, J. Multivariate Anal., 9, 3, 460-464 (1979) · Zbl 0422.62042
[5] Frangos, N. E.; Vrontos, S. D., Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance, ASTIN Bull.: J. IAA, 31, 1, 1-22 (2001) · Zbl 1035.62108
[6] Frees, E. W.; Gao, J.; Rosenberg, M., Predicting the frequency and amount of health care expenditures, N. Am. Actuar. J., 15, 3, 377-392 (2011)
[7] Frees, E. W.; Kim, J.-S., Multilevel model prediction, Psychometrika, 71, 1, 79-104 (2006) · Zbl 1306.62413
[8] Frees, E. W.; Meyers, G.; Cummings, A., Summarizing insurance scores using a Gini index, J. Amer. Statist. Assoc., 106, 495, 1085-1098 (2011) · Zbl 1229.62140
[9] Frees, E. W.; Meyers, G.; Cummings, A., Insurance ratemaking and a Gini index, J. Risk Insurance, 81, 2, 335-366 (2014)
[10] Frees, E. W.; Shi, P.; Valdez, E. A., Actuarial applications of a hierarchical insurance claims model, ASTIN Bull.: J. IAA, 39, 1, 165-197 (2009)
[11] Frees, E. W.; Valdez, E. A., Hierarchical insurance claims modeling, J. Amer. Statist. Assoc., 103, 484, 1457-1469 (2008) · Zbl 1286.62087
[12] Frees, E. W.; Young, V. R.; Luo, Y., A longitudinal data analysis interpretation of credibility models, Insurance Math. Econom., 24, 3, 229-247 (1999) · Zbl 0945.62112
[13] Garrido, J.; Genest, C.; Schulz, J., Generalized linear models for dependent frequency and severity of insurance claims, Insurance Math. Econom., 70, 205-215 (2016) · Zbl 1373.62515
[14] Gómez-Déniz, E., Bivariate credibility bonus-malus premiums distinguishing between two types of claims, Insurance Math. Econom., 70, 117-124 (2016) · Zbl 1373.62517
[15] Hausman, J. A.; Hall, B. H.; Griliches, Z., Econometric models for count data with an application to the patents-R&D relationship (1984), national bureau of economic research Cambridge: national bureau of economic research Cambridge Massachusetts, USA
[16] Hernández-Bastida, A.; Fernández-Sánchez, M.; Gómez-Déniz, E., The net Bayes premium with dependence between the risk profiles, Insurance Math. Econom., 45, 2, 247-254 (2009) · Zbl 1231.91198
[17] Jeong, H.; Ahn, J.; Park, S.; Valdez, E. A., Generalized linear mixed models for dependent compound risk models, Variance (2020), (in press)
[18] Jeong, H.; Gan, G.; Valdez, E. A., Association rules for understanding policyholder lapses, Risks, 6, 3, 69 (2018)
[19] Jeong, H.; Valdez, E. A., Bayesian credibility premium with GB2 copulas, Depend. Model., 8, 1, 157-171 (2020) · Zbl 1457.62158
[20] Kim, G.-S.; Paik, M. C.; Kim, H., Causal inference with observational data under cluster-specific non-ignorable assignment mechanism, Comput. Statist. Data Anal., 113, 88-99 (2017) · Zbl 1464.62101
[21] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models: from Data to Decisions, Vol. 715 (2012), John Wiley & Sons · Zbl 1272.62002
[22] Lee, Y.; Nelder, J. A., Hierarchical generalized linear models, J. R. Stat. Soc. Ser. B Stat. Methodol., 58, 4, 619-678 (1996) · Zbl 0880.62076
[23] Lee, W.; Park, S. C.; Ahn, J. Y., Investigating dependence between frequency and severity via simple generalized linear models, J. Korean Stat. Soc., 48, 1, 13-28 (2019) · Zbl 1411.62299
[24] Lemaire, J., Bonus-malus systems: the european and asian approach to merit-rating, N. Am. Actuar. J., 2, 1, 26-38 (1998) · Zbl 1081.91548
[25] Molenberghs, G.; Verbeke, G.; Demétrio, C.; Vieira, A., A family of generalized linear models for repeated measures with normal and conjugate random effects, Statist. Sci., 25, 3, 325-347 (2010) · Zbl 1329.62342
[26] Nelder, J.; Wedderburn, R., Generalized linear models, J. R. Statist. Soc. Ser. A (Gen.), 135, 3, 370-384 (1972)
[27] Rootzén, H.; Tajvidi, N., Multivariate generalized pareto distributions, Bernoulli, 12, 5, 917-930 (2006) · Zbl 1134.62028
[28] Shevchenko, P. V.; Wuthrich, M. V., The structural modelling of operational risk via Bayesian inference: combining loss data with expert opinions, J. Oper. Risk, 1, 3, 3-26 (2006)
[29] Shi, P.; Feng, X.; Ivantsova, A., Dependent frequency-severity modeling of insurance claims, Insurance Math. Econom., 64, 417-428 (2015) · Zbl 1348.91180
[30] Shi, P.; Valdez, E. A., Longitudinal modeling of insurance claim counts using jitters, Scand. Actuar. J., 2012, 1-21 (2012)
[31] Shi, P.; Valdez, E. A., Multivariate negative binomial models for insurance claim counts, Insurance Math. Econom., 55, 18-29 (2014) · Zbl 1296.91169
[32] Winkelmann, R., Econometric Analysis of Count Data (2008), Springer Science & Business Media
[33] Yang, X.; Frees, E. W.; Zhang, Z., A generalized beta copula with applications in modeling multivariate long-tailed data, Insurance Math. Econom., 49, 2, 265-284 (2011) · Zbl 1218.62049
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