Estimation of Periodically Changing Failure Rate

This paper considers two life testing procedures (progressively censored samples and Bartholomew's experiment) under the assumption that the life of an item follows the exponential distribution. The failure rates are different under n different conditions of usage of the item at regular intervals of time. The maximum likelihood estimates of the n failure rates have been derived along with their asymptotic variances for both types of data (when failure times are recorded and when only the number of items failing in each interval are recorded). A numerical example illustrates the type of data and relevant calculations for the experiment involving progressively censored samples.     

Reliability sampling plans for lognormal distribution, based onprogressively-censored samples

This paper presents reliability sampling plans for the lognormal distribution based on progressively censored samples. In constructing these sampling plans, large-sample approximations to the best linear unbiased estimators of the location and scale parameters are used. For some selected progressive censoring schemes, reliability sampling plans are tabulated for p_α and p_β to match MIL-STD-105. While in general, variable-sampling plans require smaller sample size when compared with attribute-sampling plans, the ordinary complete and right-censored life test experiments are special cases of the progressively censored experiment. Hence, the progressively censored reliability sampling plans in this paper are widely applicable. General application of the procedure is discussed, and two examples are provided     

Reliability Sampling Plans Under Progressive Type-I Interval Censoring Using Cost Functions

This paper gives a reliability sampling plan for progressively type I interval censored life tests when the lifetime follows the exponential distribution. We use the maximum likelihood method to obtain the point estimation of the parameter of failure time distribution. We provide an approach to establish reliability sampling plans which minimize the total cost of life testing under given consumer's and producer's risks. Some numerical studies are investigated to illustrate the proposed approach.      

Maximum likelihood estimation of Burr XII distribution parameters under Type II censoring

The mathematical theory for the point estimation of the parameters of the Burr Type XII distribution by maximum likelihood (ML) is developed for Type II censored samples. Also derived are necessary and sufficient conditions on the sample data that guarantee the existence, uniqueness and finiteness of the ML parameter estimates for all possible permissible parameter combinations. The asymptotic theory of ML is invoked to obtain approximate confidence intervals for the ML parameter estimates. An application to reliability data arising in a life test experiment is discussed.     

Testing Exponentiality Based on Kullback-Leibler Information With Progressively Type-II Censored Data

We express the joint entropy of progressively censored order statistics in terms of an incomplete integral of the hazard function, and provide a simple estimate of the joint entropy of progressively Type-II censored data. We then construct a goodness-of-fit test statistic based on Kullback-Leibler information with progressively Type-II censored data. Finally, by using Monte Carlo simulations, the power of the test is estimated, and compared against several alternatives under different progressive censoring schemes     

Estimation of scale parameters for two exponential distributions[reliability theory]

The scale parameter of the exponential distribution is estimated using conditional specification. When there are two censored samples available for estimating the scale parameter, a preliminary test is usually used to determine whether to pool the samples or to use the individual minimum-variance unbiased estimator. This latter estimator (usual preliminary-test estimator) is studied. The optimum levels of significance and their corresponding critical values for the preliminary test are obtained on the basis of the minimax regret criterion. A preliminary-test shrinkage estimator is proposed, and the optimum values of its shrinkage estimator is proposed, and the optimum values of its shrinkage coefficients are obtained. For a mean-square-error criterion of goodness of estimation, the preliminary-test shrinkage estimator is better than the usual preliminary-test estimator     

Optimum Bivariate Step-Stress Accelerated Life Test for Censored Data

A step-stress accelerated life test for two stress variables is developed. The time to failure follows the Weibull distribution, and the test is subject to termination at a predetermined time, leading to censored failure data. An optimum test plan is developed to determine the test interval for each combination of stress levels. The scale parameter of the Weibull distribution for each combination of stress levels is defined as a log linear function of the stress levels. The optimal criterion is defined to minimize the asymptotic variance of the maximum likelihood estimator of the life for a specified reliability     

Competing causes of failure and reliability tests for Weibull lifetimes under type I progressive censoring

For many high reliability products where very few items are expected to fail during the test period, testing under normal conditions is not feasible. Further, the requirement for high reliability increases the need for test procedures which yield valuable degradation and other useful information for improving product reliability. Thus in some manufacturing and other experiments, various types of failure censored and accelerated life tests are commonly employed for life testing. In this paper we discuss Type I progressively censored variable-sampling plans for Weibull lifetime distributions under competing causes of failure. The proposed procedure is attractive as it yields useful degradation-related information for improving product quality. In addition, the procedure is useful when a test is conducted under severe time constraint and/or when the experimenter wishes to save costly specimens or scarce test facilities for other use.     

Estimation of parameters of life from progressively censored data using Burr-XII model

Based on progressively Type-II censored samples, the maximum likelihood, and Bayes estimators for some lifetime parameters (reliability, and hazard functions), as well as the parameters of the Burr-XII model, are derived. The Bayes estimators are obtained using both the symmetric (Squared Error, SE) loss function, and asymmetric (LINEX, and General Entropy, GE) loss functions. This was done with respect to the conjugate prior for the one shape parameter, and discrete prior for the other parameter of this model. Also the existence, uniqueness, and finiteness of the ML parameter estimates for this type of censoring are discussed. A practical example consisting of data from an accelerated test on insulating fluid reported by Nelson (1982) was used for illustration, and comparison. Finally, some numerical results using simulation study concerning different sample sizes, and progressive censoring schemes were reported.     

A class of shrinkage estimators for the scale parameter of theexponential distribution

A class of shrinkage estimators for the scale parameter of the exponential distribution is suggested. It includes some previously published estimators as special cases. An analogous estimator based on censored samples is considered. An example is given     

Solution of the Three-Parameter Weibull Equations by Constrained Modified Quasilinearization (Progressively Censored Samples)

The location, shape, and scale parameters of the Weibull distribution are estimated from Type I progressively censored samples by the method of maximum likelihood. Nonlinear logarithmic likelihood estimating equations are derived, and the approximate asymptotic variance-covariance matrix for the maximum likelihood parameter estimates is given. The iterative procedure to solve the likelihood equations is a stable and rapidly convergent constrained modified quasilinearization algorithm which is applicable to the general case in which all three parameters are unknown. The numerical results indicate that, in terms of the number of iterations required for convergence and in the accuracy of the solution, the proposed algorithm is a very effective technique for solving systems of logarithmic likelihood equations for which all iterative approximations to the solution vector must satisfy certain intrinsic constraints on the parameters. A FORTRAN IV program implementing the maximum likelihood estimation procedure is included.     

Best linear unbiased estimator of the Rayleigh scale parameterbased on fairly large censored samples

This paper tabulates: coefficients and relative s-efficiency of the best linear unbiased estimator (BLUE) of the scale parameter of the Rayleigh distribution for type II censored samples of size N=20(5)40, r=0(1)4 (number of observations censored from the left) and s=0(1)4 (number of observations censored from the right); and ranks, coefficients, variances, and relative s-efficiencies of the BLUE of a based on a selected few order statistics (k) for sample size N=20(1)40 and k=2(1)4. These estimators have the minimum variance among the BLUE of a based on the same number of order statistics. As compared to maximum likelihood estimators (MLE) and approximate MLE, the s-efficiency of BLUE of ρ is very high. When estimating the parameter using only a few observations, the k-optimum BLUE of ρ is the only choice, as the MLE of ρ is not available. Therefore, these tables for coefficients of BLUE of ρ based on censored samples and few observations for moderately large samples, have many applications     

A weighted estimator for the scale parameter of an exponential distribution

In this paper, a weighted estimator for the scale parameter of an exponential distribution in type II censored sampling is considered, when independent samples are drawn from each of two different exponential distributions. The weight chosen is a function of two scalars, 0 ≤ a ≤ 1 and b > 0, and the test statistic, which is used for testing the equality of scale parameters of the two distributions. The mean square error of the proposed estimator is compared with that of the maximum likelihood estimator. On the basis of the relative efficiencies, the value of b is determined and then the value of a is obtained for which the mean square error of the weighted estimator is at a minimum. The proposed estimator is preferable when the two scale parameters are close to each other and the sample size is small.     

Bayes computation for life testing and reliability estimation

Powerful computational techniques for estimating the parameters and the reliability function of complex life distributions, using Bayes methods, from complete and type-II censored samples are given. The Gibbs sampler approach brings considerable conceptual and computational simplicity to the calculation of the posterior marginals and reliability. Considering constrained parameter and truncated data problems in multivariate life distributions, the Gibbs sampler procedure is easy to implement for sets of simulated data     

Bayesian shrinkage estimation of system reliability with Weibull distribution of components

This paper considers two different Bayesian shrinkage estimators of series system and parallel system reliabilities, based on type II censored samples, assuming that the lifetimes of the components follow independent Weibull distributions with known scale parameters. The mean squared errors and relative efficiencies of the estimators are investigated through Monte Carlo simulation studies.     

Approximate MLE of the scale parameter of the Rayleigh distributionwith censoring

For the Rayleigh distribution, the maximum-likelihood method does not provide an explicit estimator for the scale parameter for left-censored samples; however, such censored samples arise quite often in practice. The author provides a simple method of deriving explicit estimators by approximating the likelihood function and obtains the asymptotic variance of this estimator. He shows that this estimator is as efficient as the best linear unbiased estimator. An example is given to illustrate this method     

Estimation in a random censoring model with incomplete information:exponential lifetime distribution

Analysis of methods and simulation results for estimating the exponential mean lifetime in a random-censoring model with incomplete information are presented. The instant of an item's failure is observed if it occurs before a randomly chosen inspection time and the failure is signaled. Otherwise, the experiment is terminated at the instant of inspection during which the true state of the item is discovered. The maximum-likelihood method (MLM) is used to obtain point and interval estimates for item mean lifetime, for the exponential model. It is demonstrated, using Monte Carlo simulation, that the MLM provides positively biased estimates for the mean lifetime and that the large-sample approximation to the log-likelihood ratio produces accurate confidence intervals. The quality of the estimates is slightly influenced by the value of the probability of failure to signal. Properties of the Fisher information in the censored sample are investigated theoretically and numerically     

Weibull分布下的系统可靠性评估方法研究

在Weibull分布的定时截尾样本中,对可靠性、可靠寿命和失效率这3种参数的验前分布及形状参数的验前信息进行了分析。论述了结合熵损失函数来求得系统可靠度及寿命的Bayes点估计和置信下限,为大型系统的可靠性评估提供了一种重要的理论依据。     

Bayesian Analysis of the Weibull Process with Unknown Scale Parameter and Its Application to Acceptance Sampling

The Weibull process with unknown scale parameter is taken as a model for Bayesian decision making. The family of natural conjugate prior distributions for the scale parameter is exhibited and used in prior and posterior analysis. Preposterior analysis and several sampling schemes are then discussed. Preposterior analysis is given for an acceptance sampling problem with utility linear in the unknown mean of the Weibull process, in which the sampling scheme yields the first r failures in a life test of n items. An example is included.     

Bayesian Estimation of Parameters of Life Distributions and Reliability from Type II Censored Samples

The Bayesian approach to reliability estimation from Type II censored samples is discussed here with emphasis on obtaining natural conjugate prior distributions. The underlying sampling distribution from which the censored samples are drawn follows a generalized life model (GLM) which includes a model proposed by Epstein and Sobel, Weibull, exponential, and Rayleigh distributions as special cases. Results are given for the Type II asymptotic distribution of largest values, Pareto, and Limited distribution. The natural conjugate prior, Bayes estimate for the generalized scale parameter, posterior risk, Bayes risk and Bayes estimate of the reliability function were derived for the distributions studied. In every case the natural conjugate prior is a 2-parameter family which provides a wide range of possible prior knowledge. Conjugate diffuse priors were derived. A diffuse prior, also called a quasi-pdf, is not a pdf because its integral is not unity. It represents roughly an informationless prior state of knowledge. The proper choice of the parameter for the diffuse prior leads to maximum likelihood, classical uniform minimum-variance unbiased estimator, and an admissible biased estimator with minimum mean square error as the generalized Bayes estimate. A feature of the GLM is the increasing function g(·) with possible applications in accelerated testing. KG(·) is a s-complete s-sufficient statistic for ?, and KG(·)/m is a maximum likelihood estimate for ?. Similar results were obtained for the Pareto, Type II asymptotic distribution of extremes, Pareto (associated with Pearl-Reed growth distribution) and others.