| Invalid time or xx time have been an important problem for operators in many fields .In the survival analysis and life test,the exponential distribution and the weibull distribution is a very important distribution.We often use the exponential distribution as the model from the the research Usually taking out a simple with size n X1, X2,…,Xn from a population means getting n observesx1,x2,,xn of the sample and this sample is called complete sample. Though in the many research we can't make all the elements invalid in the life test limited by the time ,money and so on. And also we can't observe the lives of all the triers for the limit of the test time. we can not get the complete sample limited by the al kinds of conditions. Under such conditions we only get a uncomplete sample. Generally speaking only knowing a partly sure life of the observed individual and the residual life exceeding some given value, we called this Censoring Test. The censoring test have two kinds of type-one censoring test and type-two censoring test .The problem of this article is estimating the parameterθof the exponential distribution and the weibull distribution of two parameter. f(x) from the random truncation where f(t|λ) =λe-λx, (x≥0,λ> 0). the Weilull distribution of two parameter f(x) from the random truncation where f(t) =λμtrt-1e-λtμ(t> 0,λ> 0,β> 0) In the second part and forth part, we have studied asymptotically optimal emporical bayes estimation of the exponential distribution under type-one censoring test and type-two censoring test . In the third part and fifth part, we have studied asymptotically optimal emporical bayes estimation of the weibull distribution under type-one censoring test and type-two censoring test . and in the every part, We have proved the admissibility of Bayes estimation . The following theorems described our main results.Estimating the patameter of the exponential distribution under No changed type-one censoring test :Theorem 1 under loss function L(θ, d) = (λ- d)2, (0 < d < +∞), The Bayes estimatorδB =α+r/β+S1,WhichΓdistribution the prior distribution , is admissible.provide S1 =∑i=1rti+(n-r)tr.Theorem 2 Estimating the patameter of the exponential distribution under No changed type-one censoring test, under loss function L(θ, d) = (λ- d)2, (0 < d <+∞).and makingΓdistribution as the prior distribution,The estimatorδn =α+r/β+S1 is asymptotically optimal emporical bayes estimation, provideα= r2/(S2 - r),β= Tr(S2 - r), S1 =sum from i=1 toτti + (n -r)tτ.Estimating the patameter of the exponential distribution under Changed typeonecensoring test :Theorem 3 under loss function L(θ, d) = (λ- d)2, (0 < d < +∞) . The Bayes estimatorδB =α+r/β+S2,WhichΓdistribution the prior distribution , is admissible.provide S2 = ntr.Theorem 4 Estimating the patameter of the exponential distribution under Changed type-one censoring test, under loss function L(θ, d) = (λ- d)2, (0 < d <+∞) ,and makingΓdistribution as the prior distribution,The estimatorδn =α+r/β+S2 is asymptotically optimal emporical bayes estimation. provideα= r2/(S2 -r),β= Tr/(S2 - r), S2 = ntr.Estimating the patameter of the weibull distribution under type-one censoring test : Theorem 5 under loss function L(θ, d) = (λ- d)2, (0 < d < +∞) , The Bayes estimatorδB =α+r/β+t,WhichΓdistribution the prior distribution, is admissible.Theorem 6 under loss function L(θ, d) = (λ- d)2, (0 < d < +∞) ,and makingΓdistribution as the prior distribution, The estimatorδn =α+r/β+t is asymptotically optimal emporical bayes estimation. provideα=rt2-t2+2rS2/rS2+t2,β=t3+tS2/rS2-t2.Estimatingthe patameter of the exponential distribution under No changed type-two censoring test :Theorem 7 under loss function L(θ, d) = (λ- d)2, (0 < d < +∞) , The Bayes estimatorδB =α+r/β+S3,WhichΓdistribution the prior distribution ,is admissible, provide S3 =∑i=1r ti + (n-r)τ.Theorem 8 Estimating the patameter of the exponential distribution under No changed type-two censoring test, under loss function L(θ, d) = (λ- d)2, (0 < d <+∞),and makingΓdistribution as the prior distribution.The estimatorδn =α+r/β+S3 is asymptotically optimal emporical bayes estimation. provideα= r2/(S2 - r),β= Tr(S2 - r), S3 =sum from i=1 toτti + (n -r)τ.Estimating the patameter of the exponential distribution under Changed typetwocensoring test :Theorem 9 under loss function L(θ, d) = (λ- d)2, (0 < d < +∞) , The Bayes estimatorλ=α+r/β+S4,Which T distribution the prior distribution,is admissible. provide S4 = nτ.Theorem 10 Estimating the patameter of the exponential distribution under Changed type-two censoring test, under loss function L(θ, d) = (λ- d)2, (0 < d <+∞) ,and makingΓdistribution as the prior distribution,The estimatorδn =α+r/β+S4 is asymptotically optimal emporical bayes estimation, provideα= r2/(S2 -r),β= Tr/(S2 - r), S4 = nτ.Estimating the patameter of the weibull distribution under type-two censoring test :Theorem 11 under loss function L(θ, d) = (λ-d)2, (0 < d < +∞) , The Bayes estimatorδB =α+r/β+t .WhichΓdistribution the prior distribution , is admissible.Theorem 12 under loss function L(θ, d) = (λ-d)2, (0 < d < +∞) ,and makingΓdistribution as the prior distribution, The estimatorδn =α+r/β+t is asymptotically optimal emporical bayes estimation, provideα=rt2-t2+2rS2/rS2+t2,β=t3+tS2/rS2-t2.
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