A comparison of three estimators of the Weibull parameters

Using mean square error as the criterion, we compare two least squares estimates of the Weibull parameters based on non‐parametric estimates of the unreliability with the maximum likelihood estimates (MLEs). The two non‐parametric estimators are that of Herd–Johnson and one recently proposed by Zimmer. Data was generated using computer simulation with three small sample sizes (5, 10 and 15) with three multiply‐censored patterns for each sample size. Our results indicate that the MLE is a better estimator of the Weibull characteristic value, θ, than the least squares estimators considered. No firm conclusions may be made regarding the best estimate of the Weibull shape parameter, although the use of maximum likelihood is not recommended for small sample sizes. Whenever least squares estimation of both Weibull parameters is appropriate, we recommend the use of the Zimmer estimator of reliability. Copyright © 2001 John Wiley & Sons, Ltd.     

Some Percentile Estimators for Weibull Parameters

A percentile estimator for the shape parameter of the Weibull distribution, based on the 17th and 97th sample percentiles, is proposed which is asymptotically about 66% efficient when compared with the MLE (maximum likelihood estimator). A two-observation percentile estimator, based on the 40th and 82nd sample percentiles, for the scale parameter when the shape parameter is unknown is asymptotically about 82y0 efficient when compared with the MLE. The 24th and 93rd sample percentiles yield asymptotically about 41ye jointly efficient percentile estimators for both the scale and shape parameters in a class of two-observation percentile estimators when compared with their MLEs. Some other simple percentile estimators for these parameters are also briefly discussed. Finally, asymptotic properties of these estimators are investigated and their application in statistical inference problems is mentioned.     

Bias correction for the least squares estimator of Weibull shape parameter with complete and censored data

Estimation of the Weibull shape parameter is important in reliability engineering. However, commonly used methods such as the maximum likelihood estimation (MLE) and the least squares estimation (LSE) are known to be biased. Bias correction methods for MLE have been studied in the literature. This paper investigates the methods for bias correction when model parameters are estimated with LSE based on probability plot. Weibull probability plot is very simple and commonly used by practitioners and hence such a study is useful. The bias of the LS shape parameter estimator for multiple censored data is also examined. It is found that the bias can be modeled as the function of the sample size and the censoring level, and is mainly dependent on the latter. A simple bias function is introduced and bias correcting formulas are proposed for both complete and censored data. Simulation results are also presented. The bias correction methods proposed are very easy to use and they can typically reduce the bias of the LSE of the shape parameter to less than half percent.     

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.     

聚氯乙烯薄膜使用寿命预测

文中对聚氯乙烯(PVC)薄膜在实际应用环境中的使用寿命进行了快速可靠的预测。通过加速寿命试验,以断裂伸长率下降至其初始值的50%为失效指标,对PVC薄膜在户外环境因子(紫外光和热)协同作用下的老化动力学进行了研究,利用威布尔统计方程分析老化失效数据,并使用Temperature-non-thermal模型建立外推关系式,确立了PVC薄膜的寿命-环境因子统计学关系式,模拟了PVC薄膜寿命分布在环境因子(热和强紫外光)协同作用下的变化规律。最后,应用上述寿命-环境因子关系式,对PVC薄膜在上海地区的户外使用寿命进行了预测,得出150μm厚的PVC薄膜曝露于上海地区时,户外使用寿命为13个月。     

基于GM-噪声SVR方法的矿山设备可靠性参数估计

为了有效估计小子样条件下矿山设备的三参数威布尔分布可靠性模型参数,提出基于GM-噪声SVR的参数估计方法。该方法以灰色估计法(GM)为基础估计模型的位置参数,采用基于训练样本数量和噪声参数寻优的ε - 带支持向量回归机(ε-SVR)估计尺度参数和形状参数,并通过拟合的三参数威布尔分布函数分析预测和解决设备的可靠性问题。算例结果表明,GM-噪声SVR方法可以很好地用于矿山设备可靠性模型参数估计,估计某带式输送机三参数威布尔分布可靠性模型的位置参数、尺度参数和形状参数依次为3.1525、188.3763、1.0476,平均无故障时间为188 h,标准均方根误差NRMSE为0.0519。这表明该方法的可行性和有效性。     

Reliability Analysis of the CNC System Based on Field Failure Data in Operating Environments

Reliability is a measure of how well a product will perform under a certain set of conditions for a specified amount of time especially in the field environments. In this paper, a reliability study of a computer numerical control (CNC) system is described. For this analysis, field failure data from a shop manufacturing factory collected over the course of a year on approximately 20 CNC machine tools during their operating period were analyzed. Based on the field failure data, the two‐parameter exponential distribution was found to be applicable to describe the time between failures of the CNC system from among many distributions including Weibull, gamma, two‐parameter exponential, normal, and logistic using the chi‐squared test. In this paper, we discuss the reliability estimation of the CNC system based on the collected field failure data from a manufacturing factory using the maximum likelihood estimate (MLE) and uniform minimum variance estimate (UMVUE) methods. We also discuss the confidence intervals of the mean residual lifetime and reliability function. The result shows that the UMVUE method can provide much better and more accurate results in estimating the reliability of the CNC system than the MLE. This finding, on the one hand, seems to be obvious because the UMVUE is not only an unbiased estimator but also sufficient statistic with the smallest variance; on the other hand, it is not straightforward to obtain the UMVUE of any complex function, which is the reliability function in this case. This is a very important finding and is very encouraging because it indicates that the reliability analysis of the CNC system based on the UMVUE can be more than compensated by the ability of the complexity of parameter estimation method to better evaluate and predict the reliability of the CNC system. Hence, we believe that it is worth the effort to derive those parameter functions using UMVUE method. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust Regression using Probability Plots for Estimating the Weibull Shape Parameter

The Weibull shape parameter is important in reliability estimation as it characterizes the ageing property of the system. Hence, this parameter has to be estimated accurately. This paper presents a study of the efficiency of using robust regression methods over the ordinary least‐squares regression method based on a Weibull probability plot. The emphasis is on the estimation of the shape parameter of the two‐parameter Weibull distribution. Both the case of small data sets with outliers and the case of data sets with multiple‐censoring are considered. Maximum‐likelihood estimation is also compared with linear regression methods. Simulation results show that robust regression is an effective method in reducing bias and it performs well in most cases. Copyright © 2006 John Wiley & Sons, Ltd.     

A novel pseudo‐maximum likelihood estimation method for accelerated degradation tests under Wiener process

For reliability assessment based on accelerated degradation tests (ADTs), an appropriate parameter estimation method is very important because it affects the extrapolation and prediction accuracy. The well‐adopted maximum likelihood estimation (MLE) method focuses on interpolation fitting and obtains results via maximizing the likelihood of the observations. However, a best interpolation fitting does not necessarily yield a best extrapolation. In this paper, therefore, a pseudo‐MLE (P‐MLE) method is proposed to improve the prediction accuracy of constant‐stress ADTs by considering the degradation mechanism equivalence under Wiener process. In particular, the degradation mechanism equivalence is characterized by a mechanism equivalence factor which presents the proportional relationship between degradation rate and variation. Then, the mechanism equivalence factor is determined via a two‐step method. The other model parameters can be estimated by the general MLE method. The asymptotic variances of acceleration factors and the p‐quantile of product failure time under normal condition are adopted to compare the statistical properties of the proposed method and the general MLE approach. Numerical examples show that the novel P‐MLE method may not achieve a maximum likelihood but can provide more benefits regarding prediction accuracy enhancement especially when the sample size is limited.     

New EWMA control charts for monitoring the Weibull shape parameter

In this article, we first propose a new exponentially weighted moving average (EWMA ) chart for monitoring the shape parameter of the Weibull distribution. The proposed chart is developed based on the EWMA of the normal random variable, which is transformed from the easy-to-understand chi-squared random variable. In contrast, the existing EWMA charts for monitoring the shape parameter use the sample range or the unbiased estimator of the shape parameter. Unfortunately, the EWMA chart generated from sample ranges is inefficient in detecting changes due to its lack of sufficiency, whereas the one produced using unbiased estimators of the shape parameter has a highly complicated distribution that is difficult to manipulate. Simulation studies are conducted to compare the effectiveness of the proposed EWMA chart and the two existing EWMA charts. Also, a maximum likelihood estimation method is employed to estimate the change point in the process for the proposed EWMA chart once an out-of-control (OC) signal has been triggered. Further, to reduce the time for detecting the OC signal, an EWMA chart with variable sampling intervals (VSIs) for monitoring the shape parameter is developed based on the proposed EWMA chart. This EWMA chart with VSIs is studied, and its performance is evaluated. Finally, an example to demonstrate the applicability and implementation of the proposed charts is provided.     

The mean time between failures for an LCD panel

The calculation of mean time between failures is very important in reliability life data analysis. For different distributions, the values of mean time between failures are always different. The two‐parameter Weibull distribution is widely used in reliability engineering. However, some distributions may offer a better fit of data. This paper aims to develop an algorithm for determining the best‐fitted distribution of a liquid crystal display panel based on the field return data. The two‐parameter and three‐parameter Weibull distributions and other distributions such as the Burr XII distribution, the Pareto distribution and the Log‐logistic distribution are compared to provide a better characterization of the life data which is based on the maximum value of all log‐likelihood functions. We also provide a goodness‐of‐fit test for the best‐fitted distribution. It is recommended that the Burr XII distribution could be used to characterize the reliability life of a liquid crystal display panel. Copyright © 2010 John Wiley & Sons, Ltd.     

Statistical Inference From Censored Weihull Samples

In life testing experiments it is a fairly common practice to terminate the experiment before all items have failed. The Weibull distribution is often used as a model for the observations and when a computer is available maximum likelihood estimation of the parameters is to be recommended. The tables presented in this paper enable one to set confidence limits on the parameters and the reliability based on the maximum likelihood estimates for selected censoring and sample sizes.

It is also observed that, as in the case with no censoring, the maximum likelihood estimator of the reliability is very nearly unbiased and its variance is near the Cramér-Rao lower bound, Unbiasing factors for the maximum likelihood estimator of the shape parameter are given.     

Estimation of the Shape Parameter of the Weibull Distribution Using Linear Regression Methods: Non‐Censored Samples

The two‐parameter Weibull distribution is one of the most widely applied probability distributions, particularly in reliability and lifetime modelings. Correct estimation of the shape parameter of the Weibull distribution plays a central role in these areas of statistical analysis. Many different methods can be used to estimate this parameter, most of which utilize regression methods. In this paper, we presented various regression methods for estimating the Weibull shape parameter and an experimental study using classical regression methods to compare the results of the methods. A complete list of the parameter estimators considered in this study is as follows: ordinary least squares (OLS), weighted least squares (WLS, Bergman, F&T, Lu), non‐parametric robust Theil's (Theil) and weighted Theil's (WeTheil), robust Winsorized least squares (WinLS), and M‐estimators (Huber, Andrew, Tukey, Cauchy, Welsch, Hampel and Logistic). Estimator performances were compared based on bias and mean square error criteria using Monte‐Carlo simulations. The simulation results demonstrated that for small, complete, and non‐outlier data sets, the Bergman, F&T, and Lu estimators are more efficient than the others. When the data set contains one or two outliers in the X direction, Theil is the most efficient estimator. Copyright © 2012 John Wiley & Sons, Ltd.     

Parameter estimation for bivariate Weibull distribution using generalized moment method for reliability evaluation

Bivariate Weibull distribution can address the life of a system exhibiting 2‐dimensional characteristics in risk and reliability engineering. The applicability of bivariate Weibull distribution has been hindered by its difficulty with parameter estimation, as the number of parameters in bivariate Weibull distribution is more than those in univariate Weibull distribution. Considering a particular structure of a bivariate Weibull distribution model, this paper proposes a generalized moment method (GMM) for parameter estimation. This GMM method is simple, and it has proved to be efficient. The GMM can guarantee the existence and the uniqueness of the solution. A confidence interval for each estimator is derived from the moments of the bivariate distribution. The paper presents a simulation case and 2 real cases to demonstrate the proposed methods.     

Estimation and test for the parameters of the arrhenius model

Estimation and test procedures are given for the parameters of the Arrhenius model. The assumed failure distribution is of exponential or Weibull type with fixed and temperature independent shape parameter. The approach is based on failure terminated experiments on at least two different tempera-ture levels with pre-assigned and equal number of failures for each temperature level. Estimation consists of two steps, modified maximum likelihood estimation followed by ordinary least squares regression. Statistical tests are defined on acceptance regions given in the particular parameter space by circles or ellipses depending only on the experimental design.     

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.     

A Shrinkage Approach for Failure Rate Estimation of Rare Events

Systems have become more and more reliable due to technological advancement. For highly reliable systems, there are usually very few or even no failures during the testing and operation. On the other hand, given a short operating or testing time, the failure of the system is also rare even the failure rate is relatively high. The classical maximum likelihood estimation approach results in degenerated estimates zero when no failure occurs and hence meaningless. To overcome this problem, we investigate a shrinkage approach for the estimation of the failure rate of rare events from multiple heterogeneous systems provided that some of them have failures. The shrinkage estimator shrinks the MLE toward a predetermined data‐dependent point in the parameter space and could be expressed as a weighted average of MLE and the data‐dependent point. Examples are shown how the procedure can be implemented. Simulation studies show that our approach performs better than the other approaches in most cases. Copyright © 2014 John Wiley & Sons, Ltd.     

On Some Permissible Estimators of the Location Parameter of the Weibull and Certain Other Distributions

In this paper an estimator of the location parameter of the Weibull distribution is proposed which is independent of its scale and shape parameters. Several properties of this estimator are established which suggest a proper choice of three ordered sample observations insuring a permissible estimate of the location parameter. This result is valid for every distribution which has the location parameter acting as the origin or threshold parameter. Asymptotic properties of such an estimator of the location parameter of the Weibull distribution is discussed. Finally, the paper contains a brief discussion on a percentile estimator of the location parameter of the Weibull distribution and includes some numerical illustration.     

Lower Percentile Estimation of Accelerated Life Tests with Nonconstant Scale Parameter

Lower percentiles of product lifetime are useful for engineers to understand product failure, and avoiding costly product failure claims. This paper proposes a percentile re‐parameterization model to help reliability engineers obtain a better lower percentile estimation of accelerated life tests under Weibull distribution. A log transformation is made with the Weibull distribution to a smallest extreme value distribution. The location parameter of the smallest extreme value distribution is re‐parameterized by a particular 100pth percentile, and the scale parameter is assumed to be nonconstant. Maximum likelihood estimates of the model parameters are derived. The confidence intervals of the percentiles are constructed based on the parametric and nonparametric bootstrap method. An illustrative example and a simulation study are presented to show the appropriateness of the method. The simulation results show that the re‐parameterization model performs better compared with the traditional model in the estimation of lower percentiles, in terms of Relative Bias and Relative Root Mean Squared Error. Copyright © 2017 John Wiley & Sons, Ltd.     

Reliability analysis of wind turbine subassemblies based on the 3-P Weibull model via an ergodic artificial bee colony algorithm

The Weibull distribution is the most widely used model for the reliability evaluation of wind turbine subassemblies. Considering the important role of the location parameter in the three-parameter (3-P) Weibull model and its rare application in wind turbines, this study conducted a reliability analysis of wind turbine subassemblies based on field data that obeyed the 3-P Weibull distribution model via maximum likelihood estimation (MLE). An improved ergodic artificial bee colony algorithm (ErgoABC) was proposed by introducing the chaos search theory, global best solution, and Lévy flights strategy into the classical artificial bee colony (ABC) algorithm to determine the maximum likelihood estimates of the Weibull distribution parameters. This was validated against simulation calculations and proved to be efficient for high-dimensional function optimization and parameter estimation of the 3-P Weibull distribution. Finally, reliability analyses of the wind turbine subassemblies based on different types of field failure data were conducted using ErgoABC. The results show that the 3-P Weibull model can reasonably evaluate the lifetime distribution of critical wind turbine subassemblies, such as generator slip rings and main shafts, on which the location parameter has a significant effect.