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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 & Management Optimization, 2020, 16 (3) : 1311-1328. doi: 10.3934/jimo.2019004
##### References:

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##### References:
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 1.303 $\mbox{MSE}_{Z}$ 1.742 1.6159 1.5784 1.5542 1.5199 1.4986 1.3537 $\mbox{MSE}_C$ 1.8901 1.651 1.5835 1.5644 1.5487 1.533 1.5206
 $n$ 5 10 15 20 30 50 70 $\mbox{MSE}$ 1.4613 1.4145 1.3809 1.3587 1.3251 1.3113 1.303 $\mbox{MSE}_{Z}$ 1.742 1.6159 1.5784 1.5542 1.5199 1.4986 1.3537 $\mbox{MSE}_C$ 1.8901 1.651 1.5835 1.5644 1.5487 1.533 1.5206
The simulation results with w = 0:5
 $n$ 5 10 15 20 30 50 70 $\mbox{MSE}$ 1.7979 1.6422 1.617 1.5829 1.5687 1.5414 1.5238 $\mbox{MSE}_{Z}$ 1.912 1.6895 1.6355 1.6043 1.579 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 1.617 1.5829 1.5687 1.5414 1.5238 $\mbox{MSE}_{Z}$ 1.912 1.6895 1.6355 1.6043 1.579 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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