Credibility Theory Under Dependent Risks And Balanced Loss Functions

In insurance practice, one of the crucial tasks of actuaries is to determine adequate premiums to be charged for risks and there have been a variety of in-surance pricing methodologies to serve this purpose. Credibility ratemaking is a technique for predicting future claims of a risk class, given past claims of that and related risk classes. Mowbray (1914) provided an early introduction to practical aspects of this technique. Buhlmann (1967) derived a credibil-ity formula which has the form of weighted sum of the individual mean and the overall mean in a distribution-free way by using a least-squares criterion and thereby placed credibility ratemaking on a firm modeling foundation. Subsequent some important credibility model appear like Buhlmann-Straub model, Hachemeister model, Hierarchial credibility model. In fact, it has been recognized that there exist many important insurance scenarios where these classical assumptions are certainly violated and the dependence over risks are common. The Buhlmann and Biihlmann-Straub models with de-pendence structure were studied in many cases. In classical decision theory, the loss function usually focus on precision of estimation. However, goodness of fit is also a very important criterion. Thus there is a need to provide a kind of comprehensive estimation. A pioneer work Zellner (1994) introduced the notion of a balanced loss function (BLF) in the context of a general linear model to reflect both goodness of fit and precision of estimation. Credibility ratemaking under balanced loss function has studied in recently year. The credibility models for the weighted premium principle such as Esscher pre-mium principle, modified premium principle also have been discussed because they can be reduced negative safe loading caused by net premium principle. This thesis is concerned with two aspects of the credibility estimators of risk premiums which is different from classic credibility model in risk structure and loss functions, described in detail as follows.The first part of thesis includes chapters2,in which we extend the cred-ibility models with dependence structure over risks. The credibility model with dependence over risk and time. Firstly we give the relationship be-tween orthogonal projection and credibility estimator and the formulae of credibility estimator by applying orthogonal projection technique. By the orthogonal projection technique, we derive the inhomogeneous and homoge-neous credibility estimator of risk premium in Buhlmann with dependence over risk and time. The inhomogeneous credibility estimator with common effect which means the special correlation between risks is given. At last we obtained the inhomogeneous and homogeneous credibility estimator of Buhlmann-Strau with dependence over risk and time.In the second part, we focused the credibility estimator under the bal-anced loss function and weighted quadratic loss instead of being discussed under quadratic loss in classical credibility model. In chapters3, firstly the relation between the credibility premium under balanced loss function and quadratic loss function is given. The general credibility premium with depen-dence over risk and time under the balanced loss function is established. The inhomogeneous and homogeneous credibility premium with common effect is derived while some structure parameters in credibility formula is given. At last the credibility premium of exponential premium principle under balanced loss function is established.In chapters4, firstly the linear regression credibility premium with corre-lation risk under balanced loss function is derived. Particularly the credibility premium was expressed when the target estimator in balanced loss function is specialized, meanwhile the credibility linear regression model with equal cor-relation risk and common effect under balanced loss function was discussed.In chapters5, quadratic error loss in classical credibility model is re- placed by weighted balanced quadratic error loss because the missing of safety loading is caused by the loss function. Firstly a new expectation with respect to a new probability measures which is determined by weighted function and assumed probability measures is given. The credibility premium with simple Buhlmann model under some weighted balanced quadratic error loss are deduced firstly. Then the consistency of credibility premium under weighted balanced quadratic loss function is discussed. At last, based on the discussing of credibility premium weighted balanced quadratic error loss, we focused on the premium under general weighted balanced loss function LPρ,ω,δ0(θ,δ)=ωρ(δ0,δ)+(1-ω)ρ(θ,δ). The credibility premium under bal-anced entropy loss function and balanced Linex loss function are deduced by Taylor expansion.