The Bayesian Estimation And Property Of Two Populations With Partially Missing Data

Binomial distribution,Poisson distribution and the exponential distribution is very important and has a lot of practical application .On condition of complete data ,there are a lot of excellent results. In recent years, people has pay attention to the study of the missing data. In missing data types, in the form of a more important, namely each observation, the probability of a certain observations were observed. Two samples in the study, if one observation of a general observation in the control, another observer at the overall failure under the control. Literature [1] has discussed the estimation and the test of two population of Binomial distribution . literature [2]has discussed this case with two Poisson distribution of the estimated overall test literature [3] discussed the situation estimated that the overall index and the next two tests. But they only institute the maximum likelihood estimate of the overall parameters. Bayesian estimation of the parameters on the overall nature of the findings have not.This paper will focus on the three under the overall distribution of the parameters of Bayesian estimation, property and the interval estimation.What's more , through the computer simulation, the estimation results verify the excellent nature.In this paper, a structural arrangements are as follows : Section 1 : hypothetical questions.Section 2 : The Bayesian estimate of the overall discussion of two Binomial distribution and its nature.The first part,we will have Bayesian estimation of unkoown parameters of the population with missing data,and the parameter of the other overall populationλ?1 andλ? 2. And then we concludes the Bayesian estimation aboutλunder the condition ofλ₁ =λ₂=λ(unknown) And we further prove the consistency and the asy- mpotic properties aboutλ₁ andλ₂.Part II : We will have the interval estimation ofλ₁ andλthrough theorem 2.1.1 and 2.1.2 .The confidence interval aboutλ₁ is In the same way ,we will have the confidence interval aboutλwhen the confidence level is 1 -α.Part III : Through computer simulation,data shows that the relative deviation and the relative mean square error of the maximum likelihood estimation aboutλare bigger than the Baysian estimation's. That is to say the Bayesian estimation is better than the maximum likelihood estimation ,what's more ,when the missing datas are not so many,the advantage is more obvious. Section 3 : The Bayesian estimate of the overall discussion of two poisson distribution and its nature.The first part,we will have Bayesian estimation of unkoown parameters of the population with missing data,and the parameter of the other overall populationλ₁ andλ₂.And then we concludes the Bayesian estimation aboutλunder the condition ofλ₁ =λ₂=λ(unknown)And we further prove the consistency and the asy- mpotic properties aboutλ₁ andλ 2.Part II : We will have the interval estimation ofλ₁ andλthrough theorem 3.1.1 and 3.1.2 . The confidence interval aboutλ₁ isIn the same way ,we will have the confidence interval aboutλwhen the confidence level is 1-α.Part III : Through computer simulation,data shows that the relative deviation and the relative mean square error of the maximum likelihood estimation aboutλare bigger than the Baysian estimation's. That is to say the Bayesian estimation is better than the maximum likelihood estimation ,what's more ,when the missing datas are not so many,the advantage is more obvious.Section 4 : The Bayesian estimate of the overall discussion of two exponential distribution and its nature. The first part,we will have Bayesian estimation of unkoown parameters of the population with missing data,and the parameter of the other overall populationλ?1 andλ? 2. And then we concludes the Bayesian estimation aboutλunder the condition ofλ₁ =λ₂=λ(unknown)And we further prove the consistency and the asy- mpotic properties aboutλ₁ andλ 2.Part II : We will have the interval estimation ofλ₁ andλthrough theorem 4.1.1 and 4.1.2 .The confidence interval aboutλ₁ In the same way ,we will have the confidence interval aboutλwhen the confidence level is 1 -α.Part III : Through computer simulation,data shows that the relative deviation and the relative mean square error of the maximum likelihood estimation aboutλare bigger than the Baysian estimation's. That is to say the Bayesian estimation is better than the maximum likelihood estimation ,what's more ,when the missing datas are not so many,the advantage is more obvious.