Approximate Bayesian Estimation Of Continuous Loss Distribution And Its Application

Taking the uncertainty in non-life insurance as the research object,a stochastic model for the loss of insurance is established,which is the main content of non-life insurance actuarial science.Usually,in actuarial practice,economic losses are caused by insurance accidents.According to the insurance contract,the economic losses caused by the accident need to be paid by the insurance company.The object is to build a loss distribution model by the assistant of a sequence of its historical claims.The feature of a loss (3 is identified by an unknown risk parameter ,due to the heterogeneity over policies in the concerned portfolio,possible values of are described by a random variable following a probability distribution ()which is called the prior distribution in Bayesian analysis,so the estimation of falls into the Bayesian framework.A basic idea of credibility estimation is to use sample information and prior information to determine risk parameter .In classical B¨uhlmann's theory,the credibility estimation has the form of weighted sum of the individual mean and the aggregate mean.How to make full use of prior information is an important content of Bayesian statistical inference.This thesis is concerned with parameter estimation methods of two-parameter exponential distribution and lognormal distribution.Moment estimation,Bayesian estimation,maximum likelihood estimation and their statistical properties have been studied.In the case of large sample,the results of these methods are very accurate.However,in the case of small sample,Bayesian estimation methods are usually used,since Bayesian estimation depends on the form of sample distribution and prior distribution,it usually encounters the situation that the integral cannot be calculated and the display solution cannot be obtained.Therefore,based on the credibility theory,We introduce a new function class,and propose quadratic Bayesian estimation of scale parameters for two-parameter exponential distribution and lognormal distribution.Quadratic Bayesian estimation does not depend on the specific form of prior distribution,only the first four order moments of the prior distribution are needed,and it has a closed analytic solution form and is convenient to use.For different loss distributions,the expression of quadratic Bayesian estimation has a unified form.Under the mean square error criterion,the quadratic Bayesian estimation is better than the classical credibility estimation and maximum likelihood estimation.Given the form of the prior distribution,in the case of a fixed prior mean,a prior distribution with more concentrated prior information can reduce the mean square error of quadratic Bayesian estimation.From the perspective of approximating Bayesian estimation,quadratic Bayesian estimation approaching Bayesian estimation is robust to the choice of prior distribution,and the degree of approximation is better than linear Bayesian estimation.Finally,based on an actual data,the effectiveness of the quadratic Bayesian estimation is verified.