Statistical Analysis Of Reliability About Zero-Failure Data

In the reliability test, we are always getting the censored data and in the censoredⅡ test, especially for some items with high reliability and small number sampling, it always appear the zero—failure. The research on the zero—failure data has more important and more practical values with the zero—failure data appeared more in the recent years.Example: m times censoredⅡ tests are proceeding on a production, the censored time is and the samples number is ,if none of samples are failure, then it is called zero—failure data . We can remember . It means the samples number of the censored time is equal to or above , i.e. at moment , living samples number.In this paper, we discuss mainly the two questions of the zero—failure data as the follows:一,How to use zero—failure data to estimate parameters of reliability index.In article[13], a method dealing with zero—failure data, that is, distribution curves method, is given. The basis thought is: at first estimation the failure probability at moment , then approximation by least-squares method to predict the life distribution parameters and finally getting the reliability estimation. The important part of it is estimation of the failure probability . (A) Weibull distribution: . (1) In which: are both unknown.Theorem1 For the life distribution function is Weibull distribution(1), with the parameter , m times censored tests are proceeding on the production and obtain zero—failure data ,＋,,then the variable range of is satisfied: , where , , is the Bayesian estimation of . If ,the prior density function of is uniformly distribution of , then under the quadratic loss function condition, the Bayesian estimation of is as follows.Theorem1 For the life distribution function is Weibull distribution(1), with the parameter , m times censored tests are proceeding on the production and obtain zero—failure data ,＋, then the variable range of is satisfied: , in which ,, is the Bayesian estimation of , , is the up—limit of (determined by experts). If , the prior density function of is uniformly distribution of , then under the quadratic loss function condition, the Bayesian estimation of is as follows.(B) Exponential distribution: the density function is as follows , (2)in which is the failure –rate of the .We get the hierarchical natural and conjugated prior distribution of the failure—rate , which is as follows: . (3)In which , is quasi-Gamma function.Theorem2 For the life distribution function is (2), m+1 times censoredⅡ tests are proceeding on the production and the first m times ones get zero—failure data , but the (m+1)th test, with the censored time and the samples number , get one—failure data. If , the prior density function of is given in (3), then under the quadratic loss function condition, the Bayesian estimation of is as follows ,In which , , , is the estimation of under the zero—failure data, i.e. , (4)where , is quasi--Gamma function. We call the synthesize estimation of the failure—rate of the zer0—failure data .Theorem2 For the life distribution function is (2), m+1 times censoredⅡ tests are proceeding on the production and the first m times ones get zero—failure data , but the (m+1)th test, with the censored time and the samples number , get r—failure data (r=1,2,…,). If , the prior density function of is given in (3), then under the quadratic loss function condition, the Bayesian estimation of is as follows , In which ,, , is given in (4), .We call the synthesize estimation of the failure—rate of the zero—failure data .二,Reliability demonstration testing of the zero—failure dataFor the given life distribution, if we need to ascertain...