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May  2020, 16(3): 1311-1328. doi: 10.3934/jimo.2019004

Multidimensional balanced credibility model with time effect and two level random common effects

School of Science, Nanjing University of Science and Technology, Nanjing, China

* Corresponding author: zhangqiang189219@163.com

Received  October 2017 Revised  March 2018 Published  May 2020 Early access  March 2019

Fund Project: The authors are supported by NSFC grant 11271189 and Scientific Research Innovation Project of Jiangsu Province grant KYZZ116_0175

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.

Citation: Qiang Zhang, Ping Chen. Multidimensional balanced credibility model with time effect and two level random common effects. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1311-1328. doi: 10.3934/jimo.2019004
References:
[1]

C. BolancéM. 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. DeyM. 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. EbrahimzadehN. IbrahimA. 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. EnglundM. GuillénJ. GustafssonL. Hansen and J. Nielsen, Multivariate latent risk: A credibility approach, Astin Bulletin, 38 (2008), 137-146.  doi: 10.1017/S0515036100015099.

[6]

M. EnglundJ. GustafssonJ. 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. 

[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. FreesV. 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. FreesV. 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. JafariE. 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.

[16]

O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models, Belgian Actuarial Bulletin, 2 (2001), 73-79. 

[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. WenX. 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. ZhangQ. 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. ZhangL. 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. BolancéM. 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. DeyM. 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. EbrahimzadehN. IbrahimA. 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. EnglundM. GuillénJ. GustafssonL. Hansen and J. Nielsen, Multivariate latent risk: A credibility approach, Astin Bulletin, 38 (2008), 137-146.  doi: 10.1017/S0515036100015099.

[6]

M. EnglundJ. GustafssonJ. 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. 

[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. FreesV. 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. FreesV. 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. JafariE. 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.

[16]

O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models, Belgian Actuarial Bulletin, 2 (2001), 73-79. 

[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. WenX. 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. ZhangQ. 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. ZhangL. 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.

Table 1.  The simulation results with w = 0:2
$n$ 5 10 15 20 30 50 70
$\mbox{MSE}$ 1.4613 1.4145 1.3809 1.3587 1.3251 1.3113 1.3030
$\mbox{MSE}_{Z}$ 1.7420 1.6159 1.5784 1.5542 1.5199 1.4986 1.3537
$\mbox{MSE}_C$ 1.8901 1.6510 1.5835 1.5644 1.5487 1.5330 1.5206
$n$ 5 10 15 20 30 50 70
$\mbox{MSE}$ 1.4613 1.4145 1.3809 1.3587 1.3251 1.3113 1.3030
$\mbox{MSE}_{Z}$ 1.7420 1.6159 1.5784 1.5542 1.5199 1.4986 1.3537
$\mbox{MSE}_C$ 1.8901 1.6510 1.5835 1.5644 1.5487 1.5330 1.5206
Table 2.  The simulation results with w = 0:5
$n$ 5 10 15 20 30 50 70
$\mbox{MSE}$ 1.7979 1.6422 1.6170 1.5829 1.5687 1.5414 1.5238
$\mbox{MSE}_{Z}$ 1.9120 1.6895 1.6355 1.6043 1.5790 1.5659 1.5462
$\mbox{MSE}_C$ 2.0411 1.9881 1.9484 1.9284 1.8971 1.7423 1.6048
$n$ 5 10 15 20 30 50 70
$\mbox{MSE}$ 1.7979 1.6422 1.6170 1.5829 1.5687 1.5414 1.5238
$\mbox{MSE}_{Z}$ 1.9120 1.6895 1.6355 1.6043 1.5790 1.5659 1.5462
$\mbox{MSE}_C$ 2.0411 1.9881 1.9484 1.9284 1.8971 1.7423 1.6048
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