The net Bayes premium with dependence between the risk profiles. (English) Zbl 1231.91198
Summary: In Bayesian analysis it is usual to assume that the risk profiles \(\Theta _{1}\) and \(\Theta _{2}\) associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables \(\Theta _{1}\) and \(\Theta _{2}\). The degree of dependence is modelled using the Sarmanov-Lee family of distributions [O. V. Sarmanov, Dokl. Akad. Nauk SSSR 168, 32–35 (1966; Zbl 0203.20001); M.-L. Ting-Lee, Commun. Stat., Theory Methods 25, No. 6, 1207–1222 (1996; Zbl 0875.62205)], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.
MSC:
| 91B30 | Risk theory, insurance (MSC2010) |
| 62C10 | Bayesian problems; characterization of Bayes procedures |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
Software:
QRMReferences:
| [1] | Clemen, R.; Reilly, T., Correlations and copulas for decision and risk analysis, Management Science, 45, 208-224 (1999) · Zbl 1231.91166 |
| [2] | Cohen, L., Probability distributions with given multivariate marginals, Journal of Mathematics and Physics, 25, 2402-2403 (1984) · Zbl 0566.60013 |
| [3] | Duffley, M. R.; van Dorp, J. R., Risk analysis for large engineering projects: Modeling cost uncertainty for ship production activities, Journal of Engineering Valuation and Cost Analysis, 2, 285-301 (1998) |
| [4] | Drouet, D.; Kotz, S., Correlation and Dependence (2001), Imperial Press: Imperial Press Singapore · Zbl 0977.62004 |
| [5] | Farlie, D. J., The performance of some correlation coefficients for a general bivariate distribution, Biometrika, 47, 307-323 (1960) · Zbl 0102.14903 |
| [6] | Frangos, N.; Vrontos, S., Design of optimal bonus-malus systems with a frequency and severity component on an individual basis in automobile insurance, Astin Bulletin, 31, 1, 1-22 (2001) · Zbl 1035.62108 |
| [7] | Frees, E. W.; Valdez, E. A., Understanding relationships using copulas, North American Actuarial Journal, 2, 1, 1-25 (1998) · Zbl 1081.62564 |
| [8] | Härdle, W.; Kleinow, T.; Stahl, G., Applied Quantitative Finance, Theory and Computational Tools (2002), Springer, e-book · Zbl 1058.91037 |
| [9] | Hernández-Bastida, A.; Gómez-Déniz, E.; Pérez-Sánchez, J. M., Bayesian robustness of the compound Poisson distribution under bidimensional prior: An application to the collective risk model, Journal of Applied Statistics, 36, 8, 853-869 (2009) · Zbl 1473.62348 |
| [10] | Johnson, N. L.; Kemp, A. W.; Kotz, S., Univariate Discrete Distributions (2005), John Wiley: John Wiley New York · Zbl 1092.62010 |
| [11] | Klugman, S. A., Bayesian Statistics in Actuarial Science (1992), Kluwer Academic Press · Zbl 0753.62075 |
| [12] | Kotz, S.; van Dorp, R. J., A versatile bivariate distribution on a bounded domain: Another look at the product moment correlation, Journal of Applied Statistics, 29, 8, 1165-1179 (2002) · Zbl 1121.62412 |
| [13] | Kozubowski, T. J.; Panoska, A. K., A mixed bivariate distribution with exponential and geometric marginals, Journal of Statistical Planning and Inference, 134, 501-520 (2005) · Zbl 1072.62003 |
| [14] | McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative Risk Management: Concepts, Techniques and Tools (2005), Princeton University Press · Zbl 1089.91037 |
| [15] | Nadarajah, S.; Kotz, S., Compound mixed Poisson distributions I, Scandinavian Actuarial Journal, 3, 141-162 (2006) · Zbl 1143.91035 |
| [16] | Nadarajah, S.; Kotz, S., Compound mixed Poisson distributions II, Scandinavian Actuarial Journal, 3, 163-181 (2006) · Zbl 1144.91028 |
| [17] | Piergosch, W. W.; Casella, G., The existence of the First Negative Moment, The American Statistician, 39, 1, 60-62 (1985) |
| [18] | Rolski, T.; Schmidli, H.; Schmidt, V.; Teugel, J., Stochastic Processes for Insurance and Finance (1999), John Wiley and Sons · Zbl 0940.60005 |
| [19] | Sarmanov, O. V., Generalized normal correlation and two-dimensional Frechet classes, Doklady (Soviet Mathematics), 168, 596-599 (1966) · Zbl 0203.20001 |
| [20] | Shevchenko, P. V.; Wüthrich, M. V., The structural modelling of operational risk via Bayesian inference: Combining loss data with expert opinions, Journal of Operational Risk, 1, 3, 3-26 (2006) |
| [21] | Ting-Lee, M. L., Properties and applications of the Sarmanov family of bivariate distributions, Communications Statistics: Theory and Methods, 25, 6, 1207-1222 (1996) · Zbl 0875.62205 |
| [22] | Yi, W.; Bier, V., An application of copulas to accident precursor analysis, Management Science, 44, 5257-5270 (1998) · Zbl 0989.91536 |
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