Parameter Estimation Of ZZ Distribution Under Progressive Type-? Censored Data

Among the reliability tests related to parameter estimation,the censored life test is a widely used test method.And because modern industrial products have the characteristics of long life and high costs,in the process of censored life test,the testers can randomly select a part of the products from the non-failed products to remove out of the test,which is called the progressive censored life test.By this way,we can understand the characteristics of products and save costs.Reliability statistical inference are generally based on four common life distributions:exponential distribution,weibull distribution,extreme distribution and log-normal distribution.Under various tests or censored sample forms,there are many studies about the four distributions both at home and abroad.However,some storage products rarely fail before a given design life,and the proportion of failures after a design life increases significantly.And ZZ distribution can better describe the life distribution of such products.This article is based on progressive type-II censored data of ZZ distribution to study the estimation and property of parameters.The specific research contents are summarized as follows:Firstly,the maximum likelihood estimation and approximate maximum likelihood estimation are investigated of ZZ distribution under progressive type-II censored data.The likelihood function of the parameters is given by the joint density function,and the partial derivative of the logarithm is taken to obtain the nonlinear likelihood equations.The numerical solution can be obtained by the Newton-Raphson iterative method.Moreover,using Taylor expansion,we convert the nonlinear part of the likelihood equation system into a linear expression.That is,the analytical expressions of the parameters are obtained by the approximate maximum likelihood.Finally,a numerical simulation is performed by Matlab to verify and compare the two estimation methods.Secondly,the best linear unbiased estimation is discussed of ZZ distribution under progressive type-II censored data.Since the products are withdraw from the test at every failure time,the observed sample data is a sequential statistic with many possible cases.We first give the conditional best linear unbiased estimation of parameters under a sampling case,and then use the probability of each possible situation as a weight to obtain the general expression of the best linear unbiased estimate of the parameters under progressive type-II censored data.Moreover,two specific expressions of the best linear unbiased estimation for two special removed situations are given,and the number table is constructed for easy use.Finally,a numerical simulation example is presented to exhibit the feasibility of the general best linear unbiased estimation algorithm.Thirdly,the interval estimation of ZZ distribution parameters and the lower confidence limit of the reliability function are discussed under progressive type-II censored data.Based on EZ distribution,we construct the statistics whose distribution depends on the parameters only through the reliability function,and get the expression of the lower confidence limit of the reliability function after transformation.Next,by using the asymptotic theory of maximum likelihood estimation,we obtain the approximate confidence interval of the parameters.Finally,a numerical simulation is used to verify the effectiveness of the interval estimation method.