Bayesian Graduation And Spline Graduation Of Mortality Rate

In actuarial science and demography, we usually have to estimate unknown parameter using the samples which we have already known. Using traditional method of estimate, we usually can not draw a perfect conclusion, and it is just the problem, which graduation care about. The method and the technique of graduation belong to the sequence estimation , it not only based on the information of the sample but on the knowledge about the sequence which people have known before called priori information as well. Graduation method was originally proposed by DeForest (1870),and a monograph written by British actuatry London on Graduation Theory entitle 《GRADUATION:TheRevision of Estimates》 , was published inl985.Since then, much considerable effort has been made in the study of graduation, by scholars from abroad and domestic. As it wildly used in the construction of life table, population report and data processing, graduation was pay more attention on. Now moving-weighted-average graduation, Whittaker graduation, Bayesian graduation, Parameter graduation, Kernel graduation and Planar graduation, are some well-rounded method.This paper constructs a mortality table of men from eighteen to ninety-five based on the median of Brazilian insurers for the four years 1998 to 2001, in which using Bayesian graduation and Spline graduation. There are 4 parts in this paper, and part two Bayesian graduation and part three Spline graduation are primary. Part two introduce inverse Gaussian distribution first, then assume it is the priori distribution of death rate θ_x, deduce the posterior distribution of θ_x, but not get a shown expression, this problem was solved using MCMC, and chose the median of sample as the value of Bayesian graduation. In the third part of this paper, discuss the estimate of dead rate sequence, creates a loss function of square, and proves the uniqueness of the estimate. With the help of Matlab draws a better conclusion of the dead rate sequence. At last analyzed the sequence in brief.This paper can give some reference to insurance agent on construction a mortality table.