Synthetic Estimation Of The Parameter Under Exponential Distribution In The Case Of Zero-failure Data

In the reliability analysis, we always hold the fix-time censored test to study the quality index, such as the failure ratio, the reliability measurement and so on. However, with the development of science and techonology, the reliability of the products is higher and higher. The zero-failure data is incidental in the fix-time censored test especially in the good quality and small sample test. In this paper, I estimate the parameter synthetically using the methods which were given before under exponential distribution.Suppose that the life-time T of the product is of exponential distribution, the distribution function iswhere 0＜λ＜∞,λis the failure ratio,θ=1/λis the mean of T.To assess the reliability of the product, we hold the test as follows: dividing N samples into k groups, each group including n_i samples, i=1,2,…,k, and∑_i=1^k n_i=N.Suppose each group begin at time 0, and group i stop at time t_i which is fixed before. If in the whole test, we have no sample failed, we call (n_i,t_i) zero-failure data.Then, we take the note R_i= 1-P_i, (i=1,2,…,k), and R_i is the reliability measurement at time t_i.First, I estimate the reliability parameter using the quality of the exponential function with the failure information. There is some shortcoming when we use the convexity of the exponential function, for many distribution functions have the property. And memoryless is charac teristic for the exponential function.In the case of zero-failure data, suppose that the prior distribution of the failure probability P_i iswhere.Then the Bayesian estimation of R_i under the square loss function isWith the failure information, the Bayesian estimation of R_i(r) under the square loss function isDefinition 2.2 Let be thesynthetic estimation of the reliability measurement, where (?)_il is the Bayesian estimation of R_i in the case of zero-failure data;Then, I calculate the synthetic E-Bayes estimation of the mean of the life-time under entropy loss function.Let the prior distribution ofθis contrary-Γdistribution, its density function is where 0＜θ＜∞, a＞0, b＞0,Γ(a) = integral from n=0 to∞(t^a-1e^-t) dt is Gamma function, a and b are the preternatural parameters. According to the increasing-function method, we take b=b₀, 1＜a＜c. We should decide b₀ on the property of the product and the experienced life-time upper limit.Here, we havewhere 0＜θ＜∞, a is the preternatural parameter, and 1＜a＜c.Definiton 3.1 Let (?)_EB=∫_D (?)(a)·π(a) da=E[(?)(a)] be the E-Bayes estima tion ofθ, where D = {a:1＜a＜c},Ï€(a) is the density function of a on D, (?)(a) is the Bayes estimation ofθ.Theorem 3.1 If we hold the fix-time censored test k times with the product of exponential distribution, the result is that no sample fail with the data (n_i,t_i), let N=∑_i=1^k n_it_i, and the prior density function ofθis as (3.1), then we can conclude(1) the Bayes estimation ofθis (?)(a)=(b₀+N)/a under the entropy loss function;(2) if the prior distribution of preternatural parameter a is of the uniform distribution on (c, 1), the E-Bayes estimation ofθis (?)_EB=(b₀+N)/(c-1) In c.Theorem 3.2 If we hold the fix-time censored test k+1 times with the product of exponential distribution, the result is that for the first k times, we have no sample failed with the data (n_i,t_i), and for the k+1 times, we have r sample failed (r=0,1,…,n_k+1), let M=∑_i=1^k+1 n_it_i, and the prior density function ofθis as (3.1), then we can conclude(1) the Bayes estimation ofθis (?)(a)=(b₀+M)/(a+r) under the entropy loss function;(2) if the prior distribution of preternatural parameter a is of the uniform distribution on (c, 1), the E-Bayes estimation ofθis (?)_EB=(b₀+M)/(c-1) In (c+r)/(1+r). Definition 3.2 We callthe synthetic E-Bayes estimation ofθwith the zero-failure data (n_i, t_i), i=1,2,…,k in the case of exponential distribution. Where (?)_EB1 and (?)_EB2(r) are calculated by Theorem 3.1 and Theorem 3.2.r=0,1,…,n_k+1.