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

Qiang Zhang; Ping Chen

doi:10.3934/jimo.2019004

Article Contents

2020, Volume 16, Issue 3: 1311-1328. Doi: 10.3934/jimo.2019004

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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
^* Corresponding author: zhangqiang189219@163.com

Received: October 2017

Revised: March 2018

Early access: March 2019

Published: March 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 91B30, 62P05; Secondary: 97M30.

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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

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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

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