Credibility Evaluation And The Prediction Of The Tails Of The Exponential Dispersion Family

Exponential Dispersion Family is an extremely rich family of distribution.Since Exponential Dispersion Family was introduced to credibility theory byAmerican actuary Pro. Jewell. It has attracted a great of interest of a largenumber of Actuaries and scholars. Therefore, recently more and more researchesare focused on Exponential Dispersion Family.Seeking the credibility premium is an important subject in credibility theoryof the insurance. In the past decades, both Actuaries and scholars have devotedconsiderable effort to this field and have got a lot of results. They have obtainedthe credibility formula separately by Bayes method and Distribution Truncatedmethod.Estimating mean function of Exponential Dispersion Family and its trun-cated distribution predication also are important research topic in credibilitytheory. The method that studied this question is second order Bayes estimators,and this method represents a main tool in the second order optimal statisticstheory.This article is mainly divided into two chapters.In chapter 1, the main content is the question about credibility premiumestimator. In this chapter, firstly we state research background and significanceof credibility of the Exponential Dispersion Family. Secondly, we introduce thedefinition and some special properties of the Exponential Dispersion Family.Thirdly, we present two methods which calculate premium-the Bayes methodand Distribution Truncated method. Finally, we apply Bülhmann model intothe Exponential Dispersion Family, calculate the Bülhmann premium E(X_n+1|x₁,x₂…,x_n)=K/(K+nλ)m+nλ/(K+nλ)(?)and obtain the conclusion that Bayes premium is equal to Bülhmann premium by comparing three kinds of premium i.e., Bülhmann premium is great exactcredibility premium.In chapter 2, the main content is the estimator of mean function and thetruncated distribution prediction of Exponential Dispersion Family. In this chap-ter, firstly we state research background and practical significance. Secondly,based on previous research, explore the question further and estimate the meanfunction of Exponential Dispersion Family by second order Bayes estimatorsmethod. Thirdly, predicate the tail of the Exponential Dispersion Family usingthe above conclusion, obtain the probability expression that claim exceed thethreshold value T P(X_n+1＞T|x₁,x₂,…,x_n)=(?)(T|θ(μ))+(1/2)(1/K+n)â–³T(μ)+O_p(1/n)â–³_T(μ)=V_T(μ)/V(μ)-(?)(T|θ(μ))-(μ_T(μ)-μ(?)(T|θ(μ)))V′(μ)/V(μ)Finally, we predicate the tail of the transformed gamma distribution.The novelties of the paper are:1. In this paper, we apply Bülhmann model into the Exponential DispersionFamily and calculate the Bülhmann premium, we obtain the conclusion thatBülhmann premium is great exact credibility premium.2. In this paper, we apply the result of the chapter 2, attain the probabilityexpression of the tail of the transformed gamma distribution.