-
Previous Article
A survey of due-date related single-machine with two-agent scheduling problem
- JIMO Home
- This Issue
-
Next Article
Multicriteria investment problem with Savage's risk criteria: Theoretical aspects of stability and case study
Multidimensional balanced credibility model with time effect and two level random common effects
School of Science, Nanjing University of Science and Technology, Nanjing, China |
This paper extends the multidimensional credibility model under balanced loss function to account for not only certain conditional dependence over time for claim amounts but also dependence across individual risks and over portfolio risks. By means of orthogonal projection method in Hilbert space of random vectors, the inhomogeneous and homogeneous multidimensional credibility estimators are derived, which generalize some well known existing results in credibility theory. Moreover, the unbiased estimators of structural parameters are investigated. Finally, we present a numerical example to show the existence of the multidimensional credibility estimators and their difference from the existing ones.
References:
[1] |
C. Bolancé, M. Guillén and J. Pinquet,
Time-varying credibility for frequency risk models: Estimation and tests for autoregressive specifications on the random effects, Insurance: Mathematics and Economics, 33 (2003), 273-282.
doi: 10.1016/S0167-6687(03)00139-2. |
[2] |
H. Bühlmann and A. Gisler, A Course in Credibility Theory and its Application, 1nd edition, Springer-Verlag, Berlin, 2005. |
[3] |
D. Dey, M. Ghosh and W. Strawderman,
On estimation with balanced loss functions, Statistics and Probability Letters, 45 (1999), 97-101.
doi: 10.1016/S0167-7152(99)00047-4. |
[4] |
M. Ebrahimzadeh, N. Ibrahim, A. Jemain and A. Kilicman,
Claim dependence induced by common effects in Hierarchical credibility models, Communications in Statistics-Theory and Methods, 42 (2013), 3373-3400.
doi: 10.1080/03610926.2011.625487. |
[5] |
M. Englund, M. Guillén, J. Gustafsson, L. Hansen and J. Nielsen,
Multivariate latent risk: A credibility approach, Astin Bulletin, 38 (2008), 137-146.
doi: 10.1017/S0515036100015099. |
[6] |
M. Englund, J. Gustafsson, J. Nielsen and F. Thuring, Multidimensional credibility with time effects: An application to commercial business lines, The Journal of Risk and Insurance, 76 (2009), 443-453. Google Scholar |
[7] |
N. Farsipour and A. Asgharzadhe,
Estimation of a normal mean relative to balanced loss functions, Statistical Papers, 45 (2004), 279-286.
doi: 10.1007/BF02777228. |
[8] |
E. W. Frees, V. R. Young and Y. Luo,
A longitudinal data analysis interpretation of credibility models, Insurance: Mathematics and Economics, 24 (1999), 229-247.
doi: 10.1016/S0167-6687(98)00055-9. |
[9] |
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. |
[10] |
E. W. Frees and P. Wang,
Credibility using copulas, North American Actuarial Journal, 9 (2005), 31-48.
doi: 10.1080/10920277.2005.10596196. |
[11] |
E. Gómez-Déniz,
A generalization of the credibility theory obtained by using the weighted balanced loss function, Insurance: Mathematics and Economics, 42 (2008), 850-854.
doi: 10.1016/j.insmatheco.2007.09.002. |
[12] |
W. Z. Huang and X. Y. Wu,
Credibility premiums with dependence structure over risks and time horizon, Journal of Industrial and Management Optimization, 11 (2015), 365-380.
doi: 10.3934/jimo.2015.11.365. |
[13] |
W. Huang and X. Wu,
The credibility premiums with common effects obtained under balanced loss functions, Chinese Journal of Applied Probability and Statistics, 28 (2012), 203-216.
|
[14] |
M. Jafari, E. Marchand and A. Parsian,
On estimation with weighted balanced-type loss function, Statistics and Probability Letters, 76 (2006), 773-780.
doi: 10.1016/j.spl.2005.10.026. |
[15] |
W. S. Jewell, Multidimensional credibility, CRC Report, Berkeley: Operations Research Center, 1973. Google Scholar |
[16] |
O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models, Belgian Actuarial Bulletin, 2 (2001), 73-79. Google Scholar |
[17] |
O. Purcaru and M. Denuit,
Dependence in dynamic claim frequency credibility models, Astin Bulletin, 33 (2003), 23-40.
doi: 10.1017/S0515036100013283. |
[18] |
C. R. Rao and H. Toutenburg, Linear Models, 2nd edition, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4899-0024-1. |
[19] |
L. M. Wen, X. Y. 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. |
[20] |
L. M. Wen and X. Y. Wu,
The credibility estimator with general dependence structure over risks, Communications in Statistics-Theory and Methods, 40 (2011), 1893-1910.
doi: 10.1080/03610921003650440. |
[21] |
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. |
[22] |
A. Zellner, Bayesian and non-Bayesian estimation using balanced loss functions, in Statistical decision theory and related topics, Ⅴ (eds. S. S. Gupta and J. O. Berger), Springer, New York, (1994), 377–390. |
[23] |
Y. Zhang and L. M. Wen,
Multidimensional credibility models with random common effect, Journal of East China Normal university, 2010 (2010), 156-168.
|
[24] |
Q. Zhang, Q. Q. Cui and P. Chen,
Multidimensional credibility estimators with random common effects and time effects, Journal of Systems Science and Complexity, 30 (2017), 1107-1120.
doi: 10.1007/s11424-017-5268-8. |
[25] |
Q. Zhang, L. J. Wu and Q. Q. Cui,
The balanced credibility estimators with correlation risk and inflation factor, Statistical Papers, 58 (2017), 659-672.
doi: 10.1007/s00362-015-0719-6. |
show all references
References:
[1] |
C. Bolancé, M. Guillén and J. Pinquet,
Time-varying credibility for frequency risk models: Estimation and tests for autoregressive specifications on the random effects, Insurance: Mathematics and Economics, 33 (2003), 273-282.
doi: 10.1016/S0167-6687(03)00139-2. |
[2] |
H. Bühlmann and A. Gisler, A Course in Credibility Theory and its Application, 1nd edition, Springer-Verlag, Berlin, 2005. |
[3] |
D. Dey, M. Ghosh and W. Strawderman,
On estimation with balanced loss functions, Statistics and Probability Letters, 45 (1999), 97-101.
doi: 10.1016/S0167-7152(99)00047-4. |
[4] |
M. Ebrahimzadeh, N. Ibrahim, A. Jemain and A. Kilicman,
Claim dependence induced by common effects in Hierarchical credibility models, Communications in Statistics-Theory and Methods, 42 (2013), 3373-3400.
doi: 10.1080/03610926.2011.625487. |
[5] |
M. Englund, M. Guillén, J. Gustafsson, L. Hansen and J. Nielsen,
Multivariate latent risk: A credibility approach, Astin Bulletin, 38 (2008), 137-146.
doi: 10.1017/S0515036100015099. |
[6] |
M. Englund, J. Gustafsson, J. Nielsen and F. Thuring, Multidimensional credibility with time effects: An application to commercial business lines, The Journal of Risk and Insurance, 76 (2009), 443-453. Google Scholar |
[7] |
N. Farsipour and A. Asgharzadhe,
Estimation of a normal mean relative to balanced loss functions, Statistical Papers, 45 (2004), 279-286.
doi: 10.1007/BF02777228. |
[8] |
E. W. Frees, V. R. Young and Y. Luo,
A longitudinal data analysis interpretation of credibility models, Insurance: Mathematics and Economics, 24 (1999), 229-247.
doi: 10.1016/S0167-6687(98)00055-9. |
[9] |
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. |
[10] |
E. W. Frees and P. Wang,
Credibility using copulas, North American Actuarial Journal, 9 (2005), 31-48.
doi: 10.1080/10920277.2005.10596196. |
[11] |
E. Gómez-Déniz,
A generalization of the credibility theory obtained by using the weighted balanced loss function, Insurance: Mathematics and Economics, 42 (2008), 850-854.
doi: 10.1016/j.insmatheco.2007.09.002. |
[12] |
W. Z. Huang and X. Y. Wu,
Credibility premiums with dependence structure over risks and time horizon, Journal of Industrial and Management Optimization, 11 (2015), 365-380.
doi: 10.3934/jimo.2015.11.365. |
[13] |
W. Huang and X. Wu,
The credibility premiums with common effects obtained under balanced loss functions, Chinese Journal of Applied Probability and Statistics, 28 (2012), 203-216.
|
[14] |
M. Jafari, E. Marchand and A. Parsian,
On estimation with weighted balanced-type loss function, Statistics and Probability Letters, 76 (2006), 773-780.
doi: 10.1016/j.spl.2005.10.026. |
[15] |
W. S. Jewell, Multidimensional credibility, CRC Report, Berkeley: Operations Research Center, 1973. Google Scholar |
[16] |
O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models, Belgian Actuarial Bulletin, 2 (2001), 73-79. Google Scholar |
[17] |
O. Purcaru and M. Denuit,
Dependence in dynamic claim frequency credibility models, Astin Bulletin, 33 (2003), 23-40.
doi: 10.1017/S0515036100013283. |
[18] |
C. R. Rao and H. Toutenburg, Linear Models, 2nd edition, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4899-0024-1. |
[19] |
L. M. Wen, X. Y. 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. |
[20] |
L. M. Wen and X. Y. Wu,
The credibility estimator with general dependence structure over risks, Communications in Statistics-Theory and Methods, 40 (2011), 1893-1910.
doi: 10.1080/03610921003650440. |
[21] |
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. |
[22] |
A. Zellner, Bayesian and non-Bayesian estimation using balanced loss functions, in Statistical decision theory and related topics, Ⅴ (eds. S. S. Gupta and J. O. Berger), Springer, New York, (1994), 377–390. |
[23] |
Y. Zhang and L. M. Wen,
Multidimensional credibility models with random common effect, Journal of East China Normal university, 2010 (2010), 156-168.
|
[24] |
Q. Zhang, Q. Q. Cui and P. Chen,
Multidimensional credibility estimators with random common effects and time effects, Journal of Systems Science and Complexity, 30 (2017), 1107-1120.
doi: 10.1007/s11424-017-5268-8. |
[25] |
Q. Zhang, L. J. Wu and Q. Q. Cui,
The balanced credibility estimators with correlation risk and inflation factor, Statistical Papers, 58 (2017), 659-672.
doi: 10.1007/s00362-015-0719-6. |
5 | 10 | 15 | 20 | 30 | 50 | 70 | |
|
1.4613 | 1.4145 | 1.3809 | 1.3587 | 1.3251 | 1.3113 | 1.3030 |
|
1.7420 | 1.6159 | 1.5784 | 1.5542 | 1.5199 | 1.4986 | 1.3537 |
|
1.8901 | 1.6510 | 1.5835 | 1.5644 | 1.5487 | 1.5330 | 1.5206 |
5 | 10 | 15 | 20 | 30 | 50 | 70 | |
|
1.4613 | 1.4145 | 1.3809 | 1.3587 | 1.3251 | 1.3113 | 1.3030 |
|
1.7420 | 1.6159 | 1.5784 | 1.5542 | 1.5199 | 1.4986 | 1.3537 |
|
1.8901 | 1.6510 | 1.5835 | 1.5644 | 1.5487 | 1.5330 | 1.5206 |
5 | 10 | 15 | 20 | 30 | 50 | 70 | |
|
1.7979 | 1.6422 | 1.6170 | 1.5829 | 1.5687 | 1.5414 | 1.5238 |
|
1.9120 | 1.6895 | 1.6355 | 1.6043 | 1.5790 | 1.5659 | 1.5462 |
|
2.0411 | 1.9881 | 1.9484 | 1.9284 | 1.8971 | 1.7423 | 1.6048 |
5 | 10 | 15 | 20 | 30 | 50 | 70 | |
|
1.7979 | 1.6422 | 1.6170 | 1.5829 | 1.5687 | 1.5414 | 1.5238 |
|
1.9120 | 1.6895 | 1.6355 | 1.6043 | 1.5790 | 1.5659 | 1.5462 |
|
2.0411 | 1.9881 | 1.9484 | 1.9284 | 1.8971 | 1.7423 | 1.6048 |
[1] |
Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011 |
[2] |
Qing-Qing Yang, Wai-Ki Ching, Wan-Hua He, Na Song. Effect of institutional deleveraging on option valuation problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2097-2118. doi: 10.3934/jimo.2020060 |
[3] |
Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024 |
[4] |
Mehmet Duran Toksari, Emel Kizilkaya Aydogan, Berrin Atalay, Saziye Sari. Some scheduling problems with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021044 |
[5] |
Yue Qi, Xiaolin Li, Su Zhang. Optimizing 3-objective portfolio selection with equality constraints and analyzing the effect of varying constraints on the efficient sets. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1531-1556. doi: 10.3934/jimo.2020033 |
[6] |
Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 255-267. doi: 10.3934/naco.2020024 |
[7] |
Aurelia Dymek. Proximality of multidimensional $ \mathscr{B} $-free systems. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3709-3724. doi: 10.3934/dcds.2021013 |
[8] |
Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
[9] |
Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 |
[10] |
Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013 |
[11] |
Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021013 |
[12] |
Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709 |
[13] |
Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045 |
[14] |
Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018 |
[15] |
Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021, 16 (2) : 187-219. doi: 10.3934/nhm.2021004 |
[16] |
Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 |
[17] |
Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021019 |
[18] |
Davide La Torre, Simone Marsiglio, Franklin Mendivil, Fabio Privileggi. Public debt dynamics under ambiguity by means of iterated function systems on density functions. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021070 |
[19] |
Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021020 |
[20] |
Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]