Point and Interval Estimation, From One-Order Statistic, of the Location Parameter of an Extreme-Value Distribution with Known Scale Parameter and of the Scale Parameter of a Weibull Distribution with Known Shape Parameter

This paper derives a one-order statistic estimator ?mn b for the location parameter of the (first) extreme-value distribution of smallest values with cumulative distribution function F(x;u,b) = 1 - exp {-exp[(x-u)/b]} using the minimum-variance unbiased one-order statistic estimator for the scale parameter of an exponential distribution, as was done in an earlier paper for the scale parameter of a Weibull distribution. It is shown that exact confidence bounds, based on one-order statistic, can be easily derived for the location parameter of the extreme-value distribution and for the scale parameter of the Weibull distribution, using exact confidence bounds for the scale parameter of the exponential distribution. The estimator for u is shown to be b ln cmn + xmn, where xmn is the mth order statistic from an ordered sample of size n from the extreme-value distribution with scale parameter b and Cmn is the coefficient for a one-order statistic estimator of the scale parameter of an exponential distribution. Values of the factor cmn, which have previously viously been tabulated for n = 1(1)20, are given for n = 21(1)40. The ratios of the mean-square-errors of the maximum-likelihood estimators based on m order statistics to those of the one-order statistic estimators for the location parameter of the extreme-value distribution and the scale parameter of the Weibull distribution are investigated by Monte Carlo methods. The use of the table and related tables is discussed and illustrated by numerical examples.     

威布尔分布下VFD恒定应力加速寿命试验与统计分析

为了精确地估计真空荧光显示器(VFD)的可靠性寿命,节省试验测试时间,通过建立加速寿命试验模型开展了4组恒定应力加速寿命试验,采用威布尔函数描述VFD寿命分布,利用最小二乘法(LSM)估计威布尔参数,完成了试验数据的统计分析,并自行开发了寿命预测软件,确定了加速寿命方程,实现了VFD的寿命估计。数值结果表明,试验设计方案是正确可行的,VFD的寿命服从威布尔分布,其加速模型符合线性阿伦尼斯方程,每个加速应力水平下VFD的失效机理不变,精确计算出的VFD寿命对其生产厂商和技术人员具有重要的指导意义。     

Simultaneous Estimation of Location and Scale Parameters of the Weibull Distribution

This paper deals with the simultaneous estimation of the location parameter ? and the scale parameter ? of the Weibull distribution when both are unknown and the shape parameter ? is known. The best linear unbiased estimate (BLUE) (?, ?) based on a subset of k optimum ordered observations selected from the whole sample is compared with 1) Ogawa's asymptotically best linear estimate (ABLE) (?*, ?*) based on k ordered observations whose ranks are approximated by an asymptotic optimum selection, and 2) the BLUE based on the ranks in 1). Tables facilitating the computation of (?, ?) based on k = 3, 4 optimum ordered observations are provided.     

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.     

Assessing the Value of the Threshold Parameter in the Weibull Distribution Using Bayes Paradigm

The Weibull distribution represents a wide variety of situations. Usually, the distribution is considered as a two-parameter family with a scale, and a shape parameter. If, however, the given data reflect additional information in the form of a minimum guarantee, a positive value away from zero, it is better to go for a three-parameter model with the additional parameter known as the threshold. The threshold parameter is often very important, but increases the complexity of the model. Arbitrarily going for the three-parameter form is not advisable unless it is really required by the data. This article attempts to make a simulation-based Bayesian study for checking if the threshold parameter can be taken to be zero or positive in situations representing the two models. We study the compatibility of the models for the given data set. We conduct the posterior simulation in each case using Gibbs sampling.      

Monte Carlo Comparisons of Parameter Estimators of the 2-parameter Weibull Distribution

A Monte Carlo Simulation was carried out in order to compare three different estimators of the 2-parameter Weibull distribution. The estimators were the ML (maximum likelihood) estimators and two other estimator pairs suggested by Bain & Antle. The Bain-Antle estimators are better than the ML estimator for small samples (in that their bias, standard deviation, and rms error are smaller), whereas the ML estimator is superior in large samples.     

基于Weibull分布的电力设备寿命损耗预测

电力设备的寿命损耗是企业设备管理部门进行设备检修和更换的重要依据.根据设备寿命服从Weilbull分布,结合Weibull分布的特征,构建设备寿命损耗两个参数的数学模型,用于确定设备的寿命损耗程度,判定设备故障形态.应用实例得出Weibull模型的寿命损耗曲线与实际损耗曲线一致,通过曲线可判定设备在统计阶段所处的故障状态期为早期失效期,根据此模型可以进一步预测设备的最佳检修和更换时机,为企业制定设备维修政策提供支持.     

基于指数分布的加速寿命试验简论

讨论了4种用于描述加速寿命试验中失效分布参数和环境条件之间关系的失效物理模型。针对阿伦尼斯模型,探讨了加速寿命试验中的参数估计方法,构建了参数的极大似然估计(MLE)方程组,解得加速寿命试验中失效分布参数的MLE值,进而通过转化,借助于标准正态分布表获得其寿命指标的近似值,并通过一个实例介绍了其具体应用。     

k-Sample Maximum Likelihood Ratio Test for Change of Weibull Shape Parameter

A k-sample maximum likelihood ratio (MLR) test is derived to test equality of shape parameters for 2-parameter Weibull populations. The test is independent of the scale parameters, and the power depends on ratios of the shape parameters. Critical points and power calculations were obtained by Monte Carlo techniques for k = 2. The MLR test is equivalent to the MLR test of scale parameters for the extreme value distribution.     

基于威布尔分布的某半导体器件贮存寿命分布规律初探

对有机电致发光二极管（Organic Light-Emitting Diode,OLED）微型显示器件进行90℃、80℃、70℃的高温贮存试验,获得产品的失效数据。基于威布尔分布模型,采用最小二乘法进行参数估计,对失效数据分析,获得OLED微型显示器件失效分布函数。应用经典可靠性理论,计算产品在90℃、80℃、70℃的特征寿命、可靠寿命及平均故障间隔时间（Mean Time Between Failure,MTBF）。采用Arrhenius模型,依据90℃、80℃、70℃的贮存特征寿命,获得常温下产品的贮存特征寿命。分析结果表明,该方法合理、简便、有效,数据结果可以进一步应用到推导产品常温贮存寿命。     

Weibull分布下基于MAM的白光OLED寿命预测

为了得到白光有机发光二极管(OLED)寿命信息,降低试验成本,开展了三组恒定电流应力加速寿命试验。采用Weibull函数描述其寿命分布,基于图分析法(MAM)和MATLAB绘制的Weibull概率双坐标纸,描点作图并估计形状参数和尺度参数,实现了白光OLED的寿命预测。数值结果表明,白光OLED样品在各加速应力下失效机理保持不变,加速模型满足逆幂定律,精确计算的加速参数使得OLED寿命快速估算成为可能。     

Accelerated Life Test Planning With Independent Weibull Competing Risks With Known Shape Parameter

This article presents methodology for accelerated life test (ALT) planning when there are two or more failure modes, or competing risks which are dependent on one accelerating factor. It is assumed that the failure modes have respective latent (unobservable) failure times, and the minimum of these times corresponds to the product lifetime. The latent failure times are assumed to be s-independently distributed Weibull with known, common shape parameter. Expressions for the Fisher information matrix, and test plan criteria are presented. The methodology is applied to the ALT of Class-H insulation for motorettes, where temperature is the accelerating factor. Two-level, and 4:2:1 allocation test plans based on determinants, and on estimating quantiles or hazard functions, are presented. Sensitivity of optimal test plans to the specified Weibull shape parameter is also studied     

基于LSSVM的威布尔分布形状参数估计

固态介质击穿寿命特性通常用威布尔分布来描述,形状参数卢反应了固态介质的失效特征,因而需要精确估计β值.提出了在小样本情况下基于最小二乘支持向量机(LSSVM)的参数评估方法,并给出了LSSVM在MOS电容与时间有关的击穿寿命分布评估中的应用实例,并与常规的最小二乘评估方法相比,得到的结果表明LSSVM的评估精度更高(均方误差更小)、鲁棒性更好,在小样本情况下能更精确地确定威布尔分布的形状参数.     

Weibull分布下基于MLE的红外发光二极管寿命预测

为了对红外发光二极管(LED)恒定及步进应力加速寿命试验的数据进行统计分析,应用Weibull分布函数描述了其寿命分布,利用极大似然法(MLE)及其迭代流程图估计出形状参数和尺度参数,通过最小二乘法确定了红外LED加速寿命方程,对红外LED寿命是否符合威布尔分布进行了Kolmogorov-Smirnov检验,并利用自行开发的寿命预测软件计算出平均寿命和中位寿命。数值结果表明,红外LED的寿命服从Weibull分布,加速寿命方程符合逆幂定律,所估计出的红外LED的寿命对生产厂商和用户有很强的指导意义。     

恒加试验下LED照明灯的三参数Weibull分布的参数估计

三参数Weibull分布拟合LED照明灯寿命的优势较为明显,但要得到三参数Weibull分布参数较为精确的点估计较为困难。目前常用的参数估计方法有极大似然法、矩估计法、Bayes估计法等,由于其计算的方程复杂,导致软件编程繁琐,不易掌握,而且也不一定能得到参数估计。鉴于此,文章针对恒加试验提出一种简便地求解三参数Weibull分布参数估计的方法,该方法不涉及超越方程的求解问题,软件编程相当简单,且统计思想清晰。通过LED照明灯恒加试验下的几个案例数据说明方法的应用,并与已有的方法做了对比分析。     

一种威布尔寿命分布模型

给出了一种产品的长记忆寿命模型 ,推导出产品寿命的威布尔分布特性 ,对产品寿命试验的设计、分析、仿真有一定的应用价值     

Empirical Bayes Estimation in the Weibull Distribution

In part I empirical Bayes estimation procedures are introduced and employed to obtain an estimator for the unknown random scale parameter of a two-parameter Weibull distribution with known shape parameter. In part II, procedures are developed for estimating both the random scale and shape parameters. These estimators use a sequence of maximum likelihood estimates from related reliability experiments to form an empirical estimate of the appropriate unknown prior probability density function. Monte Carlo simulation is used to compare the performance of these estimators with the appropriate maximum likelihood estimator. Algorithms are presented for sequentially obtaining the reduced sample sizes required by the estimators while still providing mean squared error accuracy compatible with the use of the maximum likelihood estimators. In some cases whenever the prior pdf is a member of the Pearson family of distributions, as much as a 60% reduction in total test units is obtained. A numerical example is presented to illustrate the procedures.