Surveys On The Application Of Bayesian Method On Actuarial Study

This review is a summary to the application of Bayesian method on actuarial study: first elaborated Bayesian thought, Bayesian theory and the related knowledges, simultaneously made a brief summary to their applications. The second part takes the application of Bayesian method as a based line, introduced the outset and development of Bayesian model, and summarizes the Bayesian model. The third part describes the application of Bayesian model on insurance actuarial, and makes related explaining with examples.Bayesian method developed based on Bayesian theory, and is used for elaborating and solving statistical questions systematically. Expressed with the formula as follows:Ï€(θ)⊕Ρ( x |θ)â†'Ï€(θ|x),the mark"⊕"should be understand that is the function of Bayesian formula. Obviously, the statistical inference which obtains with the Bayesian methods , is result which obtains based on the historic information obtained prior probability and the sample information, compared with persuasive and accepted easily by the user. This paper describes some common Bayesian models, such as: Bühlmann model and Bühlmann-straub model.Bühlmann model is the most famous method on the determining of Experience Rating. It was the achievement of the Swiss actuarial scientist Bühlmann, he limited the Bayesian estimation in a linear combination within the scope of observations. This is not only advantageous for the computation, also favors the explanation. Bühlmann credibility premium is: P = (1 - Z )μ+ ZY, where Z = T /T +Ï…/a,0≤Z≤1,known as the credibility or the credibility factor. Consider the combination of policy possibly changes along with time, Bühlmann and straub propose the famous model buhlmann-straub in 1970. this model is composed of some Bühlmann single contracts model, which are independent and have the same structure function. be regarded as the promotion and the refinement of Bühlmann single contracts model. the expression of insurance premium of this model is: ,whereωt represent the risk exposure of the policy in the first t years, andωrepresent the total risk exposures ,that is to say,ω=ω₁ +ω₂ +…+ω_T.The application of Bayesian method on the credit model, the question which needs to solve is that for some specific policy holders, Y = yis known, of which , ,we want to predict a specific insurance claim or the number of the claims ,that is, predicted the valve of YT +1 . we can obtain the Bayesian insurance premium after some inference, that is, those are the expected value of the supposi- tion average. We can obtain the predictive value of unknown parameters and the next time predicted value by using of Bayesian method.Through the claim frequency and the frequency of claims model, we can predict the present insurance premium. In the frequency of claims model of posteriori experience estimates, we can obtain the most superior claim frequency isIn the frequency of claim model of the priori variables,λrepresent the risk parameters when the insurers insured. Then the correcte factor of the risk of insurance policy in the first j year is , then, is the risk parameters of the first j year. After being modified by the modify factor , obtains the most superior claim compensation rate isThrough the further actual operation, we can obtain some new data. We can change the insurance premium by using of the claim frequency correct model and premium correct model, make the premium tends to be reasonable. On the aspect of those correct models, we use the credibility factor again, it can be marked asΖ,Ζ_n = n/(n +β) ,obviously, 0 <Ζ< 1,then the correction of the frequency of premium is ,where Eq is the prior estimation of claim frequency, that is, a priori information, q? L= xis the observations of the actual claim frequency observations, that is, the message of the sample, q? Bis the result of adjustment though the latter to the former. The correction of the claim is , whereμ= E_mis the prior estimation of the average claim, X = m?Lis the average claim through the actual observation, m?B is the credibility estimation of the average claim, that is, carries on the adjustment by using of the statistics value X to the prior estimateμ.Simultaneously this paper introduced the chain ladder method and the B-F method, based on Bayesian statistical model. The application of chain ladder method is the more comprehensive method in the area of reserve estimation, it is based on the traffic triangle out of the relationship between the ratio of data to infer the value of future claims. Chain ladder model is based on the flow inside the triangle of information contained in the various incidents is estimated that the final compensation. The B-F model obtains the final indemnity prior estimate completely based on the extraneous information. In the article used Verrall and England (2005) method has constructed a Specific Bayesian model, when constructed the Bayesian model, we used the ultra negative binomial distribution, gave each line of independent progress factor, then defined each progress factor concrete a priori distribution.On the life table studies, this paper use Bayesian method to estimate the parameters of of two-parameter weibull distribution and the two-parameter Gompertz distribution. On the life table revision aspect, this paper described the Bayesian Graduation, take mas the final modify value.Finally, this paper list several examples of the specific applications of those models , in order to describe the application of Bayesian methods in insurance actuarial.