Conclusion Of Censored Data Of Exponential Distribution Under A Class Of Loss

The statistics analysis of data on all kinds of lives , xx time or invalid time have been an important problem for operators in many fields and especially in the engineering and biology field. And in the survival analysis and life test,the exponential distribution is a very important distribution. We often use the exponential distribution as the model from the the research of product life(e.g.Davis,1952;Epstein,1958) to the survival time of chronic sufferers(e.g.Feigl and Zelen 1965).Usually taking out a simple with size n X₁, X₂,…, X_n from a population means getting n observes x₁, x₂,…,x_n of the sample and this sample is called complete sample. Though in the many research of the engineering and biomedicine,we can not get the complete sample limited by the al kinds of conditions. For example we can't make all the elements invalid in the life test limited by the time .money and so on. And in the medicine test the triers of the kind of medicine may lost observations ( e.g.moving to another place)or stop using the medicine because of uncomfort. And also we can't observe the lives of all the triers for the limit of the test time. 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 truncation. Formally a observation is called right truncation on the value of L if its accurate value is unknown and only the observed value is more than or equal to L.Similarlya observation is called left truncation on the value of L if its accurate value is unknown and only the observed value is less than or equal to L. And L is called truncation time of the observed individual, The right truncation is familiar to us in the life data. So this article is about the right truncation.The truncation have such kinds as follows: (a) destiny truncation ; (b) timing truncation; (c) random truncation. The timing truncation is the especial instance of the random truncation with the degenerated distribution of Y = t₀.The problem of this article is estimating the parameterθof the exponential distribution F(x) from the random truncation whereandθis the average life. This article has two main parts.In the first part, we have studied the estimate problem in the case of destiny truncation aboutθas parameter,presented the Bayes estimator ofθand presented the MRE estimator ofθunder loss function , discuss the admissibility and inadmissibility of a class of linear estimators of the form cT(X) + d and the minimax estimator ofθ. The observed data in this model are the . The following theorems described our main results.Theorem 1 and where (i = 1,2,…, r).Suppose that there exists an eguivariant estimator d*(X) forθwith finite risk. Then a MRE estimator forθunder loss function L(θ, d) = (θ/d)^q+ (d/θ)^q - 2 (q≥1) is given byThe exact form of the MRE estimator ofθTheorem 2 Let X = (X₍₁₎,X₍₂₎,…, X((r)) ,the unique Bayes estimator ofθ, say d_B(X) ,under loss functionc if there is a d'(X) ,the Bayes risk of which is r(d' (X)) <∞, then the Bayes estimator is unique.Where the Bayes estimator ofθis [T(X) +β] forΓdistribution the prior distribution , which is ofη= 1/θabout parameterα,β.Theorem 3 The estimator cT(X) + d is admissible,provide 0≤c < c^*, d > 0.Theorem 4 The estimator cT(X) + d is admissible,provide c = c^*, d > 0 .Theorem 5 The estimator c^*T(X) is admissible.Theorem 6 If one of the conditions as follows is satusfu=ied,the estimator c^*T(X) will be inadmissible.(1) c < 0或d < 0; (2) 0 < c≠c*且d=0; (3) c > c^*且d > 0.Theorem 7 The estimator T(X) is Minimax estimator of under loss functionIn the second part, we have studied the estimate problem of randomly censored data aboutθas parameter, and presented the Bayes estimatorθ|^_n ofθ,moreover sayθ|^_n is to be asymptotic Minimax efficiency. Furthermore, the iterative logarithm law and an inequality for the rth (r≥2) mean error ofθ|^_n are also established, respectively. The observed data in this model are the pairs (Z₁,δ₁), (Z₂,δ₂),…, (Z_n,δ_n). The following theorems described our main results.Theorem 8 Let Z' = (Z₁,…, Z_n,δ₁,…,δ_n),the unique Bayes estimator ofθ, say d_B(Z') ,under loss function L(θ, d) =θ/d-lnθ/d-1 is shown byif there is a d'(Z') ,the Bayes risk of which is r(d'(Z')) <∞, then the Bayes estimator is unique.Where the Bayes estimator ofθis forΓdistribution the prior distribution , which is ofη=1/θabout parameterα,β.Theorem 9θ|^_n is asymptotic minimax efficiency. Theorem 10 Let theandTheorem 11 For any r≥2where...