An F approximation for two-parameter weibull and log-normal tolerance bounds based on possibly censored data

An approximation suggested in Mann, Schafer and Singpurwalla [18] for obtaining small-sample tolerance bounds based on possibly censored two-parameter Weibull and lognormal samples is investigated. The tolerance bounds obtained are those that effectively make most efficient use of sample data. Values based on the approximation are compared with some available exact values and shown to be in surprisingly good agreement, even in certain cases in which sample sizes are very small or censoring is extensive. Ranges over which error in the approximation is less than about 1 or 2 percent are determined. The investigation of the precision of the approximation extends results of Lawless [8], who considered large-sample maximum-likelihood estimates of parameters as the basis for approximate 95 percent Weibull tolerance bounds obtained by the general approach described in [18]. For Weibull (or extreme-value) data the approximation is particularly useful when sample sizes are moderately large (more than 25), but not large enough (well over 100 for severely censored data) for asymptotic normality of estimators to apply. For such cases simplified efficient linear estimates or maximum-likelihood estimates may be used to obtain the approximate tolerance bounds. For lognormal censored data, best linear unbiased estimates may be used, or any efficient unbiased estimators for which variances and covariances are known as functions of the square of the distribution variance.     

Statistical analysis for masked system life data from Marshall‐Olkin Weibull distribution under progressive hybrid censoring

This paper considers the statistical analysis of masked data in a series system, where the components are assumed to have Marshall‐Olkin Weibull distribution. Based on type‐I progressive hybrid censored and masked data, we derive the maximum likelihood estimates, approximate confidence intervals, and bootstrap confidence intervals of unknown parameters. As the maximum likelihood estimate does not exist for small sample size, Gibbs sampling is used to obtain the Bayesian estimates and Monte Carlo method is employed to construct the credible intervals based on Jefferys prior with partial information. Numerical simulations are performed to compare the performances of the proposed methods and one data set is analyzed.     

A note on a comparison of confidence interval techniques in truncated life tests

Several approximate procedures are available in the literature for obtaining confidence intervals for the parameter A of an exponential distribution based on time truncated samples. This paper contains the results of an empirical study comparing three of these procedures.     

Determining confidence bounds for highly reliable coherent systems based on a paucity of component failures

A computationally simple method for obtaining confidence bounds for highly reliable coherent systems, based on component tests which experience few or no failures, is given. Binomial and Type I censored exponential failure data are considered. Here unknown component unreliabilities are ordered by weighting factors, which are firstly presumed known then sensitivity of the confidence bounds to these assumed weights is examined and shown to be low.     

Estimation for a weibull accelerated life testing model

It is sometimes reasonable to assume that the lifetime distribution of an item belongs to a certain parametric family, and that actual parameter values depend upon the testing environment of the item. In the two-parameter Weibull family setting, suppose both the shape and scale parameters are expressible as functions of the testing environment. For various models of functional dependency on environment, maximum likelihood methods are used to estimate characteristics of interest at specified environmental levels. The methodology presented handles exact, censored, and grouped data. A detailed accelerated life testing analysis of stress-rupture data for Kevlar/epoxy composites is given.     

Selecting among weibull,lognormal and gamma distributions using complete and censored smaples

In a recent paper, Kent and Quesenberry [19] considered using certain optimal invariant statistics to select the best fitting member of a collection of probability distributions using complete samples of life data. In the present work extensions of this approach in two directions are given. First, selection for complete samples based on scale and shape invariant statistics is considered. Next, the selection problem for type I censored samples is considered, and both scale invariant and maximum likelihood selection procedures are studied. The two-parameter (scale and shape) Weibull, lognormal, and gamma distributions are considered and applications to real data are given. Results from a (small) comparative simulation study are presented.     

Inference from accelerated life tests using eyring type re-parameterizations

Least squares estimators of the parameters of the generalized Eyring Model are obtained by using data from censored life tests conducted at several accelerated environments. These estimators are obtained after establishing that the Gauss-Markov conditions for least squares estimation are satisfied. Confidence intervals for the hazard rate at use conditions are obtained after empirically showing that the logarithm of the estimate of the hazard rate at use conditions is approximately normally distributed. The coverage probabilities of the confidence intervals are also verified by a Monte Carlo experiment. The techniques are illustrated by an application to some real data.     

Progressive Type‐II censoring and coherent systems

A new connection between the distribution of component failure times of a coherent system and (adaptive) progressively Type‐II censored order statistics is established. Utilizing this property, we develop inferential procedures when the data is given by all component failures until system failure in two scenarios: In the case of complete information, we assume that the failed component is also observed whereas in the case of incomplete information, we have only information about the failure times but not about the components which have failed. In the first setting, we show that inferential methods for adaptive progressively Type‐II censored data can directly be applied to the problem. For incomplete information, we face the problem that the corresponding censoring plan is not observed and that the available inferential procedures depend on the knowledge of the used censoring plan. To get estimates for distributional parameters, we propose maximum likelihood estimators which can be obtained by solving the likelihood equations directly or via an Expectation‐Maximization‐algorithm type procedure. For an exponential distribution, we discuss also a linear estimator to estimate the mean. Moreover, we establish exact distributions for some estimators in the exponential case which can be used, for example, to construct exact confidence intervals. The results are illustrated by a five component bridge system. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 512–530, 2015     

广义逆指数分布元件的可靠性分析

在II型混合截尾样本下,得到了广义逆指数分布未知参数的最大似然估计。利用最大似然估计的渐近正态性构造了参数的渐近置信区间,运用Lindley's逼近方法和TierneyKadane's逼近方法计算出了参数的Bayes估计。最后,运用Monte-Carlo方法对上述估计方法结果作了模拟比较。     

混合指数分布模型的Bayes分析

针对截尾试验数据的情况,给出了二元混合指数分布模型的平均寿命和可靠性函数的严格的Ｂａｙｅｓ点估计,并运用最大熵准则给出了可靠性函数的近似的Ｂａｙｅｓ置信下限估计。     

Sample sizes for β-expectation tolerance limits which control both tails of the normal distribution

Sample sizes for tolerance limits are given on a normal distribution. β-Expectation tolerance intervals are defined and constructed to control the proportion in each tail of the normal distribution. We also require with a high probability that the proportion in each tail is close to its corresponding expected coverage. Under these conditions. we dctcrniine a minimum sample size and its accompanying tolerance limits. Graphs of minimum sample sizes for the tolerance limits are provided. The computation for minimum samplc sizes is based on the bivariate noncentral t distribution. Examples are given to illustrate the use of the graphs.     

Nonparametric bayesian lifetime data analysis using dirichlet process lognormal mixture model

We propose a nonparametric Bayesian lifetime data analysis method using the Dirichlet process mixture model with a lognormal kernel. A simulation‐based algorithm that implements the Gibbs sampling is developed to fit the Dirichlet process lognormal mixture (DPLNM) model using rightly censored failure time data. Five examples are used to illustrate the proposed method, and the DPLNM model is compared to the Dirichlet process Weibull mixture (DPWM) model. Results indicate that the DPLNM model is capable of estimating different lifetime distributions. The DPLNM model outperforms the DPWM model in all the examples, and the DPLNM model shows promising potential to be applied to analyze failure time data when an appropriate parametric model for the data cannot be specified. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Nonparametric estimation of some quantities of interest from renewal theory

In this paper, confidence intervals are given for two quantities of importance related to renewal processes. For each quantity, two confidence intervals are discussed. One confidence interval is given for general, all‐purpose use. Another confidence interval is given which is easier to compute, but not of general use. The case where data are subject to right censorship is also considered. Some numerical comparisons are made. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 638–649, 2003.     

Adaptive forecasting of the size of a work force subject to random withdrawals

Adaptive forecasting procedures are developed for predicting the size of a work force which is subject to random withdrawals. The procedures are illustrated using Marine Corps data in which four stages of service for incoming cohorts are distinguished. Using these data, three forecasting procedures—conditional maximum likelihood estimation of prediction intervals; tolerance intervals; and Bayes prediction intervals—are compared.     

三参数威布尔分布在寿命分析中的参数估计

威布尔分布在可靠性工程中已得到了广泛的应用.在对己给定的寿命试验数据进行可靠性分析与评估中,因简单易解多采用二参数威布尔分布,但参数估计会带来较大误差.对具有以损耗失效为特征的某些机械零部件,采用三参数威布尔分布进行拟合及参数估计,可以得到更高的精度,因而较二参数威布尔分布,更能反映产品可靠性的实际情况.     

Planning accelerated life tests for censored two‐parameter exponential distributions

We consider optimal test plans involving life distributions with failure‐free life, i.e., where there is an unknown threshold parameter below which no failure will occur. These distributions do not satisfy the regularity conditions and thus the usual approach of using the Fisher information matrix to obtain an optimal accelerated life testing (ALT) plan cannot be applied. In this paper, we assume that lifetime follows a two‐parameter exponential distribution and the stress‐life relationship is given by the inverse power law model. Near‐optimal test plans for constant‐stress ALT under both failure‐censoring and time‐censoring are obtained. We first obtain unbiased estimates for the parameters and give the approximate variance of these estimates for both failure‐censored and time‐censored data. Using these results, the variance for the approximate unbiased estimate of a percentile at a design stress is computed and then minimized to produce the near‐optimal plan. Finally, a numerical example is presented together with simulation results to study the accuracy of the approximate variance given by the proposed plan and show that it outperforms the equal‐allocation plan. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 169–186, 1999     

逐步增加Ⅱ型截尾样本下k/N(G)系统可靠性指标的置信区间

在逐步增加Ⅰ型截尾样本下,研究k/N(G)系统可靠性指标的区间估计问题。假设部件寿命服从指数分布,利用极大似然法和Bayes方法,首先给出了部件失效率的Bayes近似置信区间,其次推导出了系统可靠度和平均寿命的Bayes近似置信区间的计算公式。最后给出随机模拟例子,并对置信区间的精度进行了讨论。     

Some test functions for the parameters of the weibull distributions

Test functions, based on various types of censored and noncensored data, for testing several hypotheses about the location, the scale, and the shape parameters of the Weibull distributions are proposed. The exact sampling distributions of these test statistics are derived and their properties in special cases are discussed. A numerical example is considered to illustrate the application of the test functions. The results of this paper possess good possibility of wide application in view of the fact that hosts of real data arising from diverse fields of human endeavor are adequately described by the Weibull distribution.     

Estimating failure time distribution and its parameters based on intermediate data from a Wiener degradation model

Instead of measuring a Wiener degradation or performance process at predetermined time points to track degradation or performance of a product for estimating its lifetime, we propose to obtain the first‐passage times of the process over certain nonfailure thresholds. Based on only these intermediate data, we obtain the uniformly minimum variance unbiased estimator and uniformly most accurate confidence interval for the mean lifetime. For estimating the lifetime distribution function, we propose a modified maximum likelihood estimator and a new estimator and prove that, by increasing the sample size of the intermediate data, these estimators and the above‐mentioned estimator of the mean lifetime can achieve the same levels of accuracy as the estimators assuming one has failure times. Thus, our method of using only intermediate data is useful for highly reliable products when their failure times are difficult to obtain. Furthermore, we show that the proposed new estimator of the lifetime distribution function is more accurate than the standard and modified maximum likelihood estimators. We also obtain approximate confidence intervals for the lifetime distribution function and its percentiles. Finally, we use light‐emitting diodes as an example to illustrate our method and demonstrate how to validate the Wiener assumption during the testing. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008