A study of Weibull shape parameter: Properties and significance

The two-parameter Weibull distribution has been widely used for modelling the lifetime of products and components. In this paper we study the effect of the shape parameter on the failure rate and three variables of importance in the context of maintenance and reliability improvement. These variables are (i) time to failure, (ii) age at replacement based on risk and (iii) residual life. We propose a classification scheme for the distribution based on the shape parameter and discuss the application of the results.     

Weighted least‐squares estimation of the shape parameter of the Weibull distribution

The purpose of this paper is to propose the weighted least‐squares procedure for estimating the shape parameter of the Weibull distribution. Results from simulation studies illustrate the mean‐squared error of the weighted least‐squares estimator is smaller than competing procedures in all cases considered. Copyright © 2001 John Wiley & Sons, Ltd.     

Practical alternatives for estimating the failure probabilities of censored life data

For censored life data, Kapur and Lamberson and O'Connor recommend the use of Johnson's formula for non-parametric estimation of the failure distribution, F(t). The formula is used to calculate the adjusted ranks of the recorded failures, which are input into the median rank estimation equation of F(t). It is our experience that Johnson's formula is fairly difficult for the reliability practitioner to understand and implement. Fortunately, an alternative formula has been developed which is much easier to use. It is demonstrated that the calculated adjusted ranks may be used in either the mean rank or median rank equations for the estimation of F(t). The question which we pose is the following: How does the performance of Johnson's estimator compare with that of the more commonly known and understood Kaplan-Meier, or product-limit, estimator?' To answer this question, the Kaplan-Meier procedure is evaluated with respect to its equivalent adjusted rank of recorded failures. The two procedures are determined to be equivalent with respect to adjusted rank criteria. Therefore, it is proved that with Johnson's estimator adapted for use with the mean rank estimator, the two procedures will yield identical estimates of the failure probabilities. Based upon this finding, it is our recommendation thatthe reliability practitioner use the alternative formula for generation of the adjusted ranks, followed by use of either the mean or median rank formula.     

A study of two estimation approaches for parameters of Weibull distribution based on WPP

Least-squares estimation (LSE) based on Weibull probability plot (WPP) is the most basic method for estimating the Weibull parameters. The common procedure of this method is using the least-squares regression of Y on X, i.e. minimizing the sum of squares of the vertical residuals, to fit a straight line to the data points on WPP and then calculate the LS estimators. This method is known to be biased. In the existing literature the least-squares regression of X on Y, i.e. minimizing the sum of squares of the horizontal residuals, has been used by the Weibull researchers. This motivated us to carry out this comparison between the estimators of the two LS regression methods using intensive Monte Carlo simulations. Both complete and censored data are examined. Surprisingly, the result shows that LS Y on X performs better for small, complete samples, while the LS X on Y performs better in other cases in view of bias of the estimators. The two methods are also compared in terms of other model statistics. In general, when the shape parameter is less than one, LS Y on X provides a better model; otherwise, LS X on Y tends to be better.     

A new approach to MLE of Weibull distribution with interval data

We study the two-parameter maximum likelihood estimation (MLE) problem for the Weibull distribution with consideration of interval data. Without interval data, the problem can be solved easily by regular MLE methods because the restricted MLE of the scale parameter β for a given shape parameter α has an analytical form, thus α can be efficiently solved from its profile score function by traditional numerical methods. In the presence of interval data, however, the analytical form for the restricted MLE of β does not exist and directly applying regular MLE methods could be less efficient and effective. To improve efficiency and effectiveness in handling interval data in the MLE problem, a new approach is developed in this paper. The new approach combines the Weibull-to-exponential transformation technique and the equivalent failure and lifetime technique. The concept of equivalence is developed to estimate exponential failure rates from uncertain data including interval data. Since the definition of equivalent failures and lifetimes follows EM algorithms, convergence of failure rate estimation by applying equivalent failures and lifetimes is mathematically proved. The new approach is demonstrated and validated through two published examples, and its performance in different conditions is studied by Monte Carlo simulations. It indicates that the profile score function for α has only one maximum in most cases. Such good characteristic enables efficient search for the optimal value of α.     

On the validity of the Weibull failure model for brittle particles

The Weibull theory of material strength and fracture assumes that the Weibull modulus m is a material parameter, which does not depend on shape and size of the loaded object. Based on large data sets from single-particle fracture experiments with brittle materials (glass, clinker cement, limestone), the authors show that the Weibull modulus of nearly spherical particles seems to decrease with increasing particle diameter. A possible explanation is that the inner structure of the particles depends on their size so that small particles are much stronger than large ones. Received: 9 December 1999     

Composite scale modeling in the presence of censored data

A composite scale modeling approach can be used to combine several scales or variables into a single scale or variable. A typical application is to combine age and usage together to form a composite timescale model. The combined scale is expected to have better failure prediction capability than individual scales. Two typical models are the linear and multiplicative models. Their parameters are determined by minimizing the sample coefficient of variation of the composite scale. The minimum coefficient of variation is hard to apply in the presence of censored data. Another open issue is how to identify key variables when a number of variables are combined. This paper develops methods to handle these two issues. A numerical example is also included to illustrate the proposed methods.     

最小二乘法在Pt100参数修正中的应用

针对单只工业铂电阻自身电阻-温度特征参数与新的工业铂电阻检定规程中所给特征参数(CVD参数)之间存在的差异,提出了通过修正特征参数的方法来减小工业铂电阻在测温过程中由CVD方程引入的系统误差。采用4次多项式拟合工业铂电阻的特征方程,然后用奇异值分解(SVD)方法求解工业铂电阻在校准温度点构成的最小二乘方程组。通过实验结果验证修正后的特征参数更符合单只工业铂电阻自身的特征参数,提高了工业铂电阻的测温准确度。     

Some Complete and Censored Sampling Results for the Weibull or Extreme-Value Distribution

A simple, unbiased estimator, based on a censored sample, has been proposed by Rain [1] for the scale parameter of the Extreme-value distribution. This estimator was shown to have high efficiency and to be approximately distributed as a chi-square variable if substantial censoring occurs. Further small sample and asymptotic properties of this estimator are considered in this paper. The estimator is modified so that it is more applicable to the complete sample case and a close chi-square approximation is established for all cases. The estimator is also shown to be related to the maximum likelihood estimator.     

Methods for the continuous assessment of reliability of specialized equipment

This paper is concerned with the assessment of reliability for equipment which is highly specialized. Of particular interest is the analysis of field data early in the life of the fleet of such equipment. Factors affecting the data underlying assessments are outlined, and a basic statistical model for describing these data is introduced. Methods of assessment of equipment reliability, and the precision of such assessments, are discussed, examples of data from this basic model are analysed, and further examples illustrate the role of model parameters. The model may also be used to analyse incomplete data sets, and we consider the penalty incurred at different levels of incompleteness. The paper concludes with a discussion outlining further possible modifications and refinements to the basic model.     

More on the mis‐specification of the shape parameter with Weibull‐to‐exponential transformation

When lifetimes follow Weibull distribution with known shape parameter, a simple power transformation could be used to transform the data to the case of exponential distribution, which is much easier to analyze. Usually, the shape parameter cannot be known exactly and it is important to investigate the effect of mis‐specification of this parameter. In a recent article, it was suggested that the Weibull‐to‐exponential transformation approach should not be used as the confidence interval for the scale parameter has very poor statistical property. However, it would be of interest to study the use of Weibull‐to‐exponential transformation when the mean time to failure or reliability is to be estimated, which is a more common question. In this paper, the effect of mis‐specification of Weibull shape parameters on these quantities is investigated. For reliability‐related quantities such as mean time to failure, percentile lifetime and mission reliability, the Weibull‐to‐exponential transformation approach is generally acceptable. For the cases when the data are highly censored or when small tail probability is concerned, further studies are needed, but these are known to be difficult statistical problems for which there are no standard solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Confidence Interval Estimation for the Weibull and Extreme Value Distributions

This paper reviews methods of constructing confidence intervals for parameters or other characteristics of the Weibull or extreme value distribution. The conditional method of obtaining confidence intervals is stressed, with emphasis on the flexibility of the method, and on the computations which are necessary to use it.     

A study of the effects of mis‐specification of the Weibull shape parameter on confidence bounds based on the Weibull‐to‐exponential transformation

Many authors of reliability texts and papers recommend or employ the Weibull‐to‐exponential transformation for ease of use in testing hypotheses and in constructing confidence intervals and bounds on the Weibull characteristic value. In making this transformation, it is assumed that the Weibull shape parameter is fixed and known. Our research shows that if this parameter is mis‐specified by an amount as small as 0.10, very poor confidence intervals and bounds will result. Hence, since the shape parameter is seldom known with perfect accuracy, based on this and other research, the use of this transformation is not recommended. Copyright © 2000 John Wiley & Sons, Ltd.     

Bayesian Inference and Life Testing Plan for the Weibull Distribution in Presence of Progressive Censoring

This article deals with the Bayesian inference of unknown parameters of the progressively censored Weibull distribution. It is well known that for a Weibull distribution, while computing the Bayes estimates, the continuous conjugate joint prior distribution of the shape and scale parameters does not exist. In this article it is assumed that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameter has a conjugate prior distribution. As expected, when the shape parameter is unknown, the closed-form expressions of the Bayes estimators cannot be obtained. We use Lindley's approximation to compute the Bayes estimates and the Gibbs sampling procedure to calculate the credible intervals. For given priors, we also provide a methodology to compare two different censoring schemes and thus find the optimal Bayesian censoring scheme. Monte Carlo simulations are performed to observe the behavior of the proposed methods, and a data analysis is onducted for illustrative purposes.     

Upper Bounds for the Power of Invariant Tests for the Exponential Distribution with Weibull Alternatives

The problem of testing for the exponential distribution (with scale, or both location and scale parameters unknown) against Weibull alternatives is considered. Upper bounds for the power of any invariant test are presented. Tests based upon the maximum likelihood estimator of the shape parameter, or a modification of it, are given which virtually achieve these bounds.     

Estimation of mixed Weibull distribution parameters using the SCEM-UA algorithm: Application and comparison with MLE in automotive reliability analysis

The shuffled complex-evolution metropolis algorithm (SCEM-UA) is used to estimate mixed Weibull distribution parameters in automotive reliability analysis. The results are compared with maximum likelihood estimation (MLE) results. The comparison shows that, in the examples given, SCEM-UA can deliver more accurate results than MLE overall.     

基于EMD多尺度威布尔分布与HMM的轴承性能退化评估方法

轴承作为旋转机械中的重要部件,对其性能退化状态进行准确评估是开展预测性维护的重要前提。针对现有性能退化指标在鲁棒性和敏感性上的不足,提出一种基于多尺度威布尔分布与隐马尔可夫模型(Hidden Markov model,HMM)的滚动轴承性能退化评估方法。首先,采用经验模态分解(empirical mode decomposition,EMD)对轴承振动信号进行多尺度分解,将轴承振动数据分解到不同尺度的本征模态分量(intrinsic mode function,IMF)中;然后,通过峭度指标选取故障特征信息明显的IMF分量,并对各个IMF分量进行滑动窗口威布尔分布拟合,提取多尺度威布尔形状参数作为性能退化特征;最后,将轴承正常状态下退化特征参数输入隐马尔可夫模型(Hidden Markov model,HMM)进行训练,建立性能退化评估模型,从而实现轴承性能退化评估。试验结果表明,该评估方法可以有效反映轴承的性能退化趋势,与其他相关方法相比,该方法能够及时识别到轴承早期故障,并且具有较强的稳定性。     

Using the empirical moment generating function in testing for the Weibull and the type I extreme value distributions

We introduce two families of statistics, functionals of the empirical moment generating function process of the logarithmically transformed data, for testing goodness of fit to the two-parameter Weibull distribution or, equivalently, to the type I extreme value model. We show that when affine invariant estimators are used for the parameters of the extreme value distribution, the distributions of these statistics to not depend on the underlying parameters and one of them has a limiting chi-squared distribution. We estimate, via simulations, some finite sample quantiles for the statistics introduced and evaluate their power against a varied set of alternatives.     

Prediction of stress and displacement fields in stress concentration regions by least squares asymptotic analysis of displacement data

This paper demonstrates that highly accurate predictions of the stress fields, including the peak stress of a stress concentration region, can be obtained easily by a least squares asymptotic analysis (LSAA) of even a relatively sparse set of displacement data from points in this zone lying sufficiently far from the boundary to avoid 'edge effects'.