# American Institute of Mathematical Sciences

April  2015, 11(2): 365-380. doi: 10.3934/jimo.2015.11.365

## Credibility models with dependence structure over risks and time horizon

 1 Department of Mathematics, Shanghai Maritime University, Shanghai, China 2 Center of International Finance and Risk Management, Department of Statistics and Actuarial Science, East China Normal University, Shanghai, China

Received  November 2012 Revised  March 2014 Published  September 2014

In this paper, the Bühlmann and Bühlmann-Straub's credibility models with a type of dependence structures over risks and over time are discussed. The inhomogeneous and homogeneous credibility estimators of risk premium were derived. The inhomogeneous credibility estimators for the existing credibility models with common effects are extended to slightly more general versions. The results obtained shake the classical meaning of the term credibility premiums''.
Citation: Weizhong Huang, Xianyi Wu. Credibility models with dependence structure over risks and time horizon. Journal of Industrial and Management Optimization, 2015, 11 (2) : 365-380. doi: 10.3934/jimo.2015.11.365
##### References:
 [1] C. Bolancé, M. Guillén, M. Denuit and J. Pinquet, Bonus-malus scales in segmented tariffs with stochastic migration between segments, Insurance: Mathematics and Economics, 33 (2003), 273-282. doi: 10.1016/S0167-6687(03)00139-2. [2] H. Bühlmann, Experience rating and credibility, Astin Bulletin, 4 (1967), 199-207. [3] H. Bühlmann and E. Straub, Glaubwüdigkeit für Schadensäze, Bulletin of the Swiss Association of Actuaries, 70 (1970), 111-133. [4] H. Bühlmann and A. Gisler, A Course in Credibility Theory and its Applications, Springer, Netherlands, 2005. [5] D. R. Dannenburg, Crossed classification credibility models, Transactions of the 25th International Congress of Actuaries, 4 (1995), 1-35. [6] J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Theory, Insurance: Mathematics and Economics, 31 (2002), 3-33. doi: 10.1016/S0167-6687(02)00134-8. [7] J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Applications, Insurance: Mathematics and Economics, 31 (2002), 133-161. doi: 10.1016/S0167-6687(02)00135-X. [8] J. Dhaene and M. J. Goovaerts, Dependency of risks and stop-loss order, Astin Bulletin, 26 (1996), 201-212. [9] E. W. Frees, V. R. Young and Y. Luo, A Longitudinal Date Analysis Interpretation of Credibility models, Insurance: Mathematics and Economics, 24 (1999), 229-247. doi: 10.1016/S0167-6687(98)00055-9. [10] E. W. Frees, V. R. Young and Y. Luo, Case studies using panel data models, North American Actuarial Journal, 5 (2001), 24-42. doi: 10.1080/10920277.2001.10596010. [11] C. A. Hachemeister, Credibility for regression models with application to trend, In Credibility, theory and application. Proceedings of the Berkeley Actuarial Research Conference on credibility, Academic Press, New York, (1975), 129-169. [12] W. S. Jewell, The use of collateral data in credibility theory: A hierarchical model, Giorndle dell'lstituto Italianodegdi Attuari, 38 (1975), 1-16. [13] T. Y. Lu and Y. Zhang, Generalized correlation order and stop-loss order, Insurance: mathematics and economics, 35 (2004), 69-76. doi: 10.1016/j.insmatheco.2004.04.003. [14] A. Müller, Stop-loss order for portfolios of dependent risks, Insurance: Mathematics and Economics, 21 (1997), 219-223. doi: 10.1016/S0167-6687(97)00032-2. [15] M. Pan, R. Wang and X. Wu, On the consistency of credibility premiums regarding Esscher principle, Insurance: Mathematics and Economics, 42 (2008), 119-126. doi: 10.1016/j.insmatheco.2007.01.009. [16] O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models, Belgian Actuarial Bulletin, 2 (2002), 73-79. [17] O. Purcaru and M. Denuit, Dependence in dynamic claim frequency credibility models, Astin Bulletin, 33 (2003), 23-40. doi: 10.2143/AST.33.1.1037. [18] S. S. Wang, V. R. Young and H. H. Panjer, Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics, 21 (1997), 173-183. doi: 10.1016/S0167-6687(97)00031-0. [19] L. Wen, X. Wu and X. Zhao, The credibility premiums under generalized weighted loss functions, Journal of Industrial and Management Optimization, 5 (2009), 893-910. doi: 10.3934/jimo.2009.5.893. [20] L. Wen, X. Wu and X. Zhou, The credibility premiums for models with dependence induced by common effects, Insurance: Mathematics and Economics, 44 (2009), 19-25. doi: 10.1016/j.insmatheco.2008.09.005. [21] X. Wu and X. Zhou, A new characterization of distortion premiums via countable additivity for comonotonic risks, Insurance: Mathematics and Economics, 38 (2006), 324-334. doi: 10.1016/j.insmatheco.2005.09.002. [22] K. L. Yeo and E. A. Valdez, Claim dependence with common effects in credibility models, Insurance: Mathematics and Economics, 38 (2006), 609-629. doi: 10.1016/j.insmatheco.2005.12.006.

show all references

##### References:
 [1] C. Bolancé, M. Guillén, M. Denuit and J. Pinquet, Bonus-malus scales in segmented tariffs with stochastic migration between segments, Insurance: Mathematics and Economics, 33 (2003), 273-282. doi: 10.1016/S0167-6687(03)00139-2. [2] H. Bühlmann, Experience rating and credibility, Astin Bulletin, 4 (1967), 199-207. [3] H. Bühlmann and E. Straub, Glaubwüdigkeit für Schadensäze, Bulletin of the Swiss Association of Actuaries, 70 (1970), 111-133. [4] H. Bühlmann and A. Gisler, A Course in Credibility Theory and its Applications, Springer, Netherlands, 2005. [5] D. R. Dannenburg, Crossed classification credibility models, Transactions of the 25th International Congress of Actuaries, 4 (1995), 1-35. [6] J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Theory, Insurance: Mathematics and Economics, 31 (2002), 3-33. doi: 10.1016/S0167-6687(02)00134-8. [7] J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Applications, Insurance: Mathematics and Economics, 31 (2002), 133-161. doi: 10.1016/S0167-6687(02)00135-X. [8] J. Dhaene and M. J. Goovaerts, Dependency of risks and stop-loss order, Astin Bulletin, 26 (1996), 201-212. [9] E. W. Frees, V. R. Young and Y. Luo, A Longitudinal Date Analysis Interpretation of Credibility models, Insurance: Mathematics and Economics, 24 (1999), 229-247. doi: 10.1016/S0167-6687(98)00055-9. [10] E. W. Frees, V. R. Young and Y. Luo, Case studies using panel data models, North American Actuarial Journal, 5 (2001), 24-42. doi: 10.1080/10920277.2001.10596010. [11] C. A. Hachemeister, Credibility for regression models with application to trend, In Credibility, theory and application. Proceedings of the Berkeley Actuarial Research Conference on credibility, Academic Press, New York, (1975), 129-169. [12] W. S. Jewell, The use of collateral data in credibility theory: A hierarchical model, Giorndle dell'lstituto Italianodegdi Attuari, 38 (1975), 1-16. [13] T. Y. Lu and Y. Zhang, Generalized correlation order and stop-loss order, Insurance: mathematics and economics, 35 (2004), 69-76. doi: 10.1016/j.insmatheco.2004.04.003. [14] A. Müller, Stop-loss order for portfolios of dependent risks, Insurance: Mathematics and Economics, 21 (1997), 219-223. doi: 10.1016/S0167-6687(97)00032-2. [15] M. Pan, R. Wang and X. Wu, On the consistency of credibility premiums regarding Esscher principle, Insurance: Mathematics and Economics, 42 (2008), 119-126. doi: 10.1016/j.insmatheco.2007.01.009. [16] O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models, Belgian Actuarial Bulletin, 2 (2002), 73-79. [17] O. Purcaru and M. Denuit, Dependence in dynamic claim frequency credibility models, Astin Bulletin, 33 (2003), 23-40. doi: 10.2143/AST.33.1.1037. [18] S. S. Wang, V. R. Young and H. H. Panjer, Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics, 21 (1997), 173-183. doi: 10.1016/S0167-6687(97)00031-0. [19] L. Wen, X. Wu and X. Zhao, The credibility premiums under generalized weighted loss functions, Journal of Industrial and Management Optimization, 5 (2009), 893-910. doi: 10.3934/jimo.2009.5.893. [20] L. Wen, X. Wu and X. Zhou, The credibility premiums for models with dependence induced by common effects, Insurance: Mathematics and Economics, 44 (2009), 19-25. doi: 10.1016/j.insmatheco.2008.09.005. [21] X. Wu and X. Zhou, A new characterization of distortion premiums via countable additivity for comonotonic risks, Insurance: Mathematics and Economics, 38 (2006), 324-334. doi: 10.1016/j.insmatheco.2005.09.002. [22] K. L. Yeo and E. A. Valdez, Claim dependence with common effects in credibility models, Insurance: Mathematics and Economics, 38 (2006), 609-629. doi: 10.1016/j.insmatheco.2005.12.006.
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