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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

Qiang Zhang , and  Ping Chen

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.

Keywords: Multidimensional credibility, common effect, time effect, balanced loss function, orthogonal projection.
Mathematics Subject Classification: Primary: 91B30, 62P05; Secondary: 97M30.
Citation: Qiang Zhang, Ping Chen. Multidimensional balanced credibility model with time effect and two level random common effects. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1311-1328. doi: 10.3934/jimo.2019004
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.  Google Scholar

[2]

H. Bühlmann and A. Gisler, A Course in Credibility Theory and its Application, 1nd edition, Springer-Verlag, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. W. Frees and P. Wang, Credibility using copulas, North American Actuarial Journal, 9 (2005), 31-48.  doi: 10.1080/10920277.2005.10596196.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

C. R. Rao and H. Toutenburg, Linear Models, 2nd edition, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4899-0024-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Y. Zhang and L. M. Wen, Multidimensional credibility models with random common effect, Journal of East China Normal university, 2010 (2010), 156-168.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

H. Bühlmann and A. Gisler, A Course in Credibility Theory and its Application, 1nd edition, Springer-Verlag, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. W. Frees and P. Wang, Credibility using copulas, North American Actuarial Journal, 9 (2005), 31-48.  doi: 10.1080/10920277.2005.10596196.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

C. R. Rao and H. Toutenburg, Linear Models, 2nd edition, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4899-0024-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Y. Zhang and L. M. Wen, Multidimensional credibility models with random common effect, Journal of East China Normal university, 2010 (2010), 156-168.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
Table Options

Download as excel

$n$ 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
Table Options

Download as excel

$n$ 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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