Applied Mathematical Sciences, Vol. 8, 2014, no. 161, 8045-8050 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410803 The Credibility Estimators with Dependence Over Risks Qiang Zhang College of Sciences, Shihezi University, Shihezi 832000, People s Republic of China Qianqian Cui Department of Applied Mathematics Nanjing University of Science and Technology Nanjing, 210094, People s Republic of China Lijun Wu College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, People s Republic of China Copyright c 2014 Qiang Zhang, Qianqian Cui and Lijun Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In classical credibility theory, claims are assumed to be independent over risks. However, in many practical situations, the assumptions may be violated in some situations. Hence, this paper investigates the credibility premiums with a special dependence structure among the individual risks under squared loss function. To be specific, the inhomogeneous and homogeneous credibility premiums with dependence over risks are derived for Bühlmann-Straub credibility models. Mathematics Subject Classification: 91B30, 62P05 Keywords: credibility estimator, dependence structure, individual risks, squared loss function
8046 Qiang Zhang, Qianqian Cui and Lijun Wu 1 Introduction In insurance practice, credibility theory has been widely used in commercial property of liability insurance and group health or life insurance. The wellknown credibility formulas obtained are written as a weighted sum of the average experience of the policyholder and the average of the entire collection of policyholders. These formulas are easy to understand and simple to apply due to their linear properties. The modern credibility theory is believed to be attributed to the remarkable contribution by Bühlmann (1967), which is the first one that based the credibility theory on modern Bayes statistics. For the recent detailed introduction, see Bühlmann and Gisler (2005), which describes modern credibility theory comprehensively. In classical credibility theory, the risks in a portfolio are assumed to be independent over risks. While such independence assumptions may be at least approximately appropriate in some practical situation, it is far from a universal structure in this complex world. For example, Yeo and Valdez (2006) introduced a rather special credibility model with claim dependence characterized by common effects, they derived the credibility premium under normally distributed claim amounts. Wen (2009) established the credibility estimator for the general credibility models with common effects and investigated the credibility premium under distribution-free cases. Wen and Deng (2011) rebuilt the credibility estimator for Bühlmann credibility models under an equal correlation structure over risks. Kolve and Paiva (2008) introduced the application of equal correlation in actuarial science and considered the joint PGF of the equal correlation claims and derived some desired properties. Huang (2012) rebuilt the credibility model under balanced loss function with common effects, the Bühlmann and Bühlmann-Straub credibility estimators are derived. Inspired by these papers, we will establish the credibility model with different distributions induced by a special dependent over risks. The rest of the paper is arranged as follows. In Section 2, model assumptions are introduced and some preliminaries are discussed. Section 3 derives the inhomogeneous and homogeneous credibility estimators for Bühlmann-Straub credibility models. 2 Model Assumptions and Preliminaries Consider a portfolio of K insured individuals. In this portfolio, each individual i is associated with a claim experience X ij over n i time periods j = 1, 2,, n i. Write X i = (X i1,, X ini ), i = 1, 2,, K. In the classical credibility theory, we assumed the risk quality of an individual i can be characterized by a risk parameter Θ i, which is an unobservable random variable. Given Θ i, the claims X i1,, X in, X i,n+1 are independent and identically distributed. Formally, the
The credibility estimators with dependence over risks 8047 assumptions of the model will be outlined are stated as following. Assumption 2.1 For fixed contract i, given Θ i, X ij are conditionally independent, with E(X ij Θ i ) = γ j µ(θ i ) and Var(X ij Θ i ) = γ2 j σ2 (Θ i ) m ij, where m ij are known weights. Assumption 2.2 The risk parameter Θ 1, Θ 2,, Θ K are dependent with the same structure distribution function π(θ). In addition, we assumption E(µ(Θ i )) = µ,, E(σ 2 (Θ i )) = σ 2, and Cov(µ(Θ i ), µ(θ j )) = ρ i ρ j, i j. Our interest is to predict the future claim X i,n+1 for each individual, taking into account all observed claims experience X 1, X 2, X K. So we need to solve the following optimal problem min E[(X c 0 R,c i R n i,ni +1 c 0 i c ix i ) 2 ], (2.1) In addition to those assumptions, we introduce the following notations for later use: m i = 1 n i n i j=1 m ij, X m i = 1 n i m i n i j=1 m ij γ j X ij, X = 1 n i m i ρ 2 i n i m i ρ 2 i X m i. Lemma 2.1 Under the assumption 2.1-2.2, we have (1) The means of X i and Xare given by E(X i ) = µt i, i = 1, 2,, K, E(X) = µt (2.2), where T i = (γ 1,, γ ni ) and T = (T 1,, T K). (2) The covariance between X i,ni +1 and X is given by X i,ni +1X = Cov(X i,n i +1, X) = γ ni +1ρ i (ρ 1 T 1,, ρ K T K), (2.3) (3) The covariance of Y is given by XX = diag(λ 1,, Λ K ) + (ρ 1 T 1,, ρ K T K )(ρ 1 T 1,, ρ K T K), (2.4) where Λ i = diag( γ2 i1 σ2 m i1,, γ2 in i σ 2 m ini ). (4) The inverse of the variance matrix of X is given by 1 XX = diag(λ 1 1,, Λ 1 σ 2 K ) (ρ σ 2 + K 1 W 1,, ρ K W K)(ρ 1 W 1,, ρ K W K ), (2.5) n i m i ρ 2 i where W i = ( m i1 γ 1,, m in i γ ni ) and Λ 1 i = diag( m i1,, m in i ). γi1 2 σ2 γin 2 σ 2 i
8048 Qiang Zhang, Qianqian Cui and Lijun Wu 3 The Credibility Estimators In this section, we proceed to drive the credibility estimators of X i,ni +1. The inhomogeneous credibility premiums are stated in the following theorem. Theorem 3.1 Under assumptions 2.1-2.2, the inhomogeneous credibility estimators of X i,n+i+1 for i = 1,, K are given by X i,ni +1 = γni +1[Z i X + (1 Z i )µ]. (3.1) where Z i = (ρ i ρ i n i m i )/(σ 2 + K n i m i ρ 2 i ), which is called credibility factor. Proof. In terms of Lemma 2.2 (see Wen (2009)), The inhomogeneous credibility estimator of X i,ni +1 denoted by X i,ni +1 = proj(xi,ni +1 L(Y, 1)), therefore X i,ni +1 = E(Xi,ni +1) + 1 (X E(X)), (3.2) X i,ni+1x XX By a simple calculation, we have X i,ni +1X 1 = γ ni +1ρ i σ 2 XX σ 2 + K n i m i ρ 2 i (ρ 1 W 1,, ρ K W K ), (3.3) Therefore, the theorem follows from the following computation: ρ i ρ i n i m i ρ i ρ i n i m i X i,ni +1 = γni +1 X + γ σ 2 + K i,ni +1(1 )µ. (3.4) n i m i ρ 2 i σ 2 + K n i m i ρ 2 i Remark 1 Theorem 1 builts the inhomogeneous estimators with different distribution. In our model, if we assume that γ i1 = = γ ni +1 = 1, n 1 = = n i = n and m ij = 1, then X m i = 1 nj=1 X n ij, X = ( K ρ 2 i X m i )/( K ρ 2 i ), and the credibility estimator of X i,ni +1 is n ρ i ρ i X ij X j=1 i,ni +1 = σ 2 + K ρ 2 i ρ i ρ i + (1 σ 2 + K ρ 2 i )µ. (3.5) When µ is unknown, we resort to establish the homogeneous credibility estimator of X i,ni +1.
The credibility estimators with dependence over risks 8049 Theorem 3.2 Under assumptions 2.1-2.2, the homogeneous credibility estimators of X i,ni +1 are given by X i,ni +1hom = γni +1X. (3.6) Proof. By Lemma 2.2 (see, Wen(2009)), we can obtain X i,ni +1hom = γni +1Z i X + γ ni +1(1 Z i )proj(µ Le(Y )). Since proj(µ Le(X) = µe(x ) 1 XX X E(X ) 1 XX E(X), (3.7) and inserting (2.1) and (2.4) into (3.1), we get proj(µ Le(Y ) = µ2 T 1 µ 2 T 1 XX X XX T = X. (3.8) Acknowledgements. The work was supported the National Natural Science Foundation of P.R. China [11361058]. References [1] H. Bühlmann, Experience rating and credibility, J. Astin Bulletin, 4 (1967), 199-207. [2] H. Bühlmann, A. Gisler, A Course in Credibility Theory and its Application, Springer, Netherlands, 2005. http://dx.doi.org/10.1007/3-540- 29273-x [3] W. Huang, X. Wu, The credibility premiums with common effects obtained under balanced loss functions, Chinese Journal of Applied Probability and Statistics, 28 (2012), 203-216. [4] N. Kolev, D. Paiva, Random sums of exchangeable variables and actuarial applications. Insurance: Mathematics and Economics, 1 (2008),147-153. http://dx.doi.org/10.1016/j.insmatheco.2007.01.010 [5] R. Rao, H. Toutenburg, Linear Models. Springer, New York, 1995. http://dx.doi.org/10.1007/978-1-4899-0024-1 [6] L. Wen, X. Wu, X. Zhou, The credibility premiums for models with dependence induced by common effects. Insurance: Mathematics and Economics, 1 (2009), 19-25. http://dx.doi.org/10.1016/j.insmatheco.2008.09.005
8050 Qiang Zhang, Qianqian Cui and Lijun Wu [7] L. Wen, W. Deng, The credibility models with equal correlation risks. Journal of Systems Science and Complexity, 3 (2011), 532-539. http://dx.doi.org/10.1007/s11424-010-8328-x [8] K. L. Yeo, E. A. Valdez, Claim dependence with common effects in credibility models. Insurance: Mathematics and Economics, 3 (2006), 609-629. http://dx.doi.org/10.1016/j.insmatheco.2005.12.006 Received: October 15, 2014; Published: November 19, 2014