Life Tests with Periodic Change in the Scale Parameter of a Weibull Distribution

Two life testing procedures, namely, the progressively censored samples and Bartholomew's experiment are discussed under the assumption that the life of an item follows a specialized Weibull distribution. The scale parameter is different under two different conditions of usage of the item at regular intervals of time, the shape parameter remains unchanged throughout the experiment. The maximum likelihood estimates of the two scale parameters have been derived along with their variances. 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     

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.     

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.      

Estimation of Reliability from Multiple Independent Grouped Censored Samples with Failure Times Known

Failure times of one type aircraft-engine component were recorded. In addition, life times are periodically recorded for unfailed engine components. The data are considered as multiple s-independent grouped censored samples with failure times known. The assumed failure model is the 2-parameter Weibull distribution. Maximum likelihood estimates are derived. The exponential model is used for comparison. Monte Carlo simulation is used to derive s-bias and mean square error of the estimates. The asymptotic covariance matrix was computed for the sampling conditions studied. The maximum likelihood estimates of the reliability were obtained as a function of component operating time since overhaul.     

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     

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     

Predicting from Early Failures the Last Failure Time of a (Log) Normal Sample

This paper presents a table of coefficients for best linear unbiased predictors (BLUPs) using the earliest r failure times to predict the n-th (last) failure time of a sample from a s-normal (or lognormal) life distribution. They are used to predict how long a sample of components will run until all fail, that is, how long a life test will last. This includes prediction of the life of a parallel system of n identical components from the first r component failure times. The table covers n = 2(1)10, r = 2(1)n - 1. The BLUPs are exact (approximate) for failure (time) censored data.     

An Empirical Procedure for Estimating the Parameters of a Mixed Exponential Life Testing Model

A failure-time distribution made up of two exponential distributions having two distinct failure rates and a specific mixture proportion is considered. The associated experiment is terminated after a predetermined time has elapsed and the numbers of failures in each time interval are recorded. As soon as an item fails it can be attributed to the appropriate subpopulation. An empirical procedure is proposed to estimate the 3 parameters of the model. The maximum likelihood estimators and our estimators are compared using both real and simulated data. The procedure can be extended to a mixed model of three or more exponential distributions, and can be applied to ungrouped and/or uncensored samples.     

Analysis of grouped and censored data from step-stress life test

This paper studies statistical analysis of grouped and censored data obtained from a step-stress accelerated life test. We assume that the stress change times in the step-stress life test are fixed and the lifetimes observed are type I censored. Maximum likelihood estimates and asymptotic confidence intervals for model parameters are obtained. We provide an asymptotic statistical test for the cumulative exposure model based on the grouped and type I censored data. We also present the optimum test plan for a simple step-stress test when the lifetime under constant stress is assumed exponential. Finally we give an application of our methods by applying our analysis process to a real life data set. The proposed statistical methodology is especially useful when intermittent inspection is the only feasible way of checking the status of test units during a step-stress test.     

Accelerated life tests analyzed by a piecewise exponentialdistribution via generalized linear models

Efficient industrial experiments for the reliability analysis of manufactured products consist of subjecting the units to accelerated life tests where, for each pre-fixed stress level, the experiment ends after the failure of a certain pre-fixed proportion of units or a certain test time is reached. This paper estimates the mean life of the units under usual working conditions, based on censored data obtained from units under stress conditions. This problem is approached through a generalized linear model and related inferential techniques, considering the very flexible class of failure distributions, piecewise exponential model and a log-linear stress-response relationship. The general framework has as particular cases, among others, the power law model, the Arrhenius model and the generalized Eyring model. A numerical example illustrates the methodology     

Shrunken estimators of Weibull shape parameter from type-IIcensored samples

A shrunken estimator of the Weibull shape parameter for failure censored samples is suggested. This shrunken estimator is compared with shrunken estimators previously given. Even for sample sizes of five and ten this shrunken estimator, based on failure data censored at only three and four, respectively, can be used with advantage when one has a reasonable guess for the shape parameter. This estimator has higher efficiency and nearness than other estimators     

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.     

Bayes Nonparametric Estimation of Time-dependent Failure Rate

This paper discusses Bayes nonparametric estimation of time-dependent failure rates. A point and an interval estimate of the quantiles of the failure process are given for both complete and censored samples. The prior degree-of-belief about the failure rate is expressed in the form of a hypothetical sample. A numerical example is discussed.     

On maximum likelihood estimation for the two parameter Weibull distribution

This paper examines recent results presented on maximum likelihood estimation for the two parameter Weibull distribution. In particular, we seek to explain some recently reported values for estimator bias when the data for analysis contains both times to failure and censored times in operation; our discussion centres on the generation of sample data sets. We conclude that, under appropriate conditions, estimators are asymptotically unbiased, with relatively low bias in small to moderate samples. We then present the results of some further experiments which suggest that the previously reported values for estimator bias can be attributed to the method of generating sample data sets in simulation experiments.     

An engineering approach to Bayes estimation for the Weibull distribution

In this paper an engineering approach to Bayes reliability analysis of Weibull failure data collected under a randomly censored sampling is proposed. The posterior distribution of several decision variables, such as the meanlife, the reliability function, the reliable life, and the hazard rate, are derived, when either a prior information on the reliability or a prior information on the hazard rate is available. Point estimates of the selected decision variables are given, by assuming both symmetric and asymmetric loss functions. Finally, numerical examples are presented to illustrate the proposed estimation procedures.     

融合非线性加速退化模型与失效率模型的产品寿命预测方法

针对高可靠性产品寿命数据少、获取成本高的问题,基于充分利用产品在研制、加速试验等不同环境下的退化数据、失效数据等可靠性数据的思想,提出了一种融合非线性加速退化模型和失效率模型的产品寿命预测方法.首先,根据退化数据对非线性退化过程进行分析,估计退化过程的参数;然后,根据加速退化数据及相应的加速退化模型估计加速退化模型的参数,从而得到退化参数与应力之间的关系.进一步,利用比例风险模型融合产品的寿命数据和未失效截尾数据,并基于此计算产品的可靠度函数、预测产品的寿命.实例应用验证了所提方法的有效性,同时说明了所提方法的应用价值.     

Linear Estimation of the Scale Parameter of the First Asymptotic Distribution of Extreme Values

A Lagrange multiplier technique is used to obtain linear, minimum-variance, unbiased estimators for the scale parameters of the first asymptotic distributions of smallest and largest values with known mode. Coefficients for multiplying ordered observations are computed for complete and censored samples of size n = 1(1) 15. Each sample of size n is censored from above and all m-order-statistic estimators are obtained where m ? n. Then the smallest subset of # order statistics from the set of m available order statistics is found which yields a 99% efficiency relative to the m-order-statistic estimator. The Cramér-Rao lower bound for the variances of the estimators for complete samples is derived and tabled for n = 1(1) 15. For censored samples the asymptotic variances of the maximum-likelihood m-order-statistic estimators are presented for comparative purposes.     

Censored life-data analysis using linguistic variables

Parameter estimates for censored life data using ordinary censoring methods (noninformative censoring) can be biased when the cause of censoring for a unit is related to its final life (informative censoring). An algorithm is presented for obtaining estimated lifetimes for informatively censored units. It is based on linguistic variable concepts from fuzzy set theory. These estimates are combined with the lifetimes for actual field units to form a complete sample for parameter estimation purposes     

Hazard Versus Renewal Rate of Electronic Items

Two different indexes, the hazard rate and the renewal rate, which are implied by conventional uses of the bathtub-shaped curve, are often noted in reliability. The hazard rate is applicable for a single failure time of each item, such as that of a nonrepairable part; the renewal rate is applicable for multiple failure times of each item, such as those of repairable equipment. Occasionally, remarks are made in the literature concerning the mathematical models for the bathtub-shaped hazard rate but not for the renewal rate. Furthermore, bathtub-shaped hazard and renewal curves as conventionally used are each based on certain assumptions concerning failure time distributions. Little data have been recorded for electronic parts and equipment which would substantiate the widespread use of the conventional implications of the bathtub-shaped hazard and renewal rates. The validity of the assumptions concerning the underlying distributions of failure times affects the accuracy of the results of reliability analyses, such as prediction, data analysis, formal assurance tests, operational planning, and maintenance planning. A study of the applications-oriented literature suggests that the distinction between the hazard rate and the renewal rate, as well as some associated implications, are not generally appreciated. Thus the existing situation is apt to lead engineers astray as well as others with application interests. Basic concepts and definitions are emphasized and extensions and implications are sketched. References are selected and noted for those interested in further pursuit.