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.     

A study of time‐between‐events control chart for the monitoring of regularly maintained systems

Owing to usage, environment and aging, the condition of a system deteriorates over time. Regular maintenance is often conducted to restore its condition and to prevent failures from occurring. In this kind of a situation, the process is considered to be stable, thus statistical process control charts can be used to monitor the process. The monitoring can help in making a decision on whether further maintenance is worthwhile or whether the system has deteriorated to a state where regular maintenance is no longer effective. When modeling a deteriorating system, lifetime distributions with increasing failure rate are more appropriate. However, for a regularly maintained system, the failure time distribution can be approximated by the exponential distribution with an average failure rate that depends on the maintenance interval. In this paper, we adopt a modification for a time‐between‐events control chart, i.e. the exponential chart for monitoring the failure process of a maintained Weibull distributed system. We study the effect of changes on the scale parameter of the Weibull distribution while the shape parameter remains at the same level on the sensitivity of the exponential chart. This paper illustrates an approach of integrating maintenance decision with statistical process monitoring methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Two‐Stage Reliability Sampling Plans with Bogey Tests

This article considers the design of two‐stage reliability test plans. In the first stage, a bogey test was performed, which will allow the user to demonstrate reliability at a high confidence level. If the lots pass the bogey test, the reliability sampling test is applied to the lots in the second stage. The purpose of the proposed sampling plan was to test the mean time to failure of the product as well as the minimum reliability at bogey. Under the assumption that the lifetime distribution follows Weibull distribution and the shape parameter is known, the two‐stage reliability sampling plans with bogey tests are developed and the tables for users are constructed. An illustrative example is given, and the effects of errors in estimates of a Weibull shape parameter are investigated. A comparison of the proposed two‐stage test with corresponding bogey and one‐stage tests was also performed. Copyright © 2012 John Wiley & Sons, Ltd.     

Cumulative Sum Control Charts for Monitoring Weibull‐distributed Time Between Events

The Weibull distribution can be used to effectively model many different failure mechanisms due to its inherent flexibility through the appropriate selection of a shape and a scale parameter. In this paper, we evaluate and compare the performance of three cumulative sum (CUSUM) control charts to monitor Weibull‐distributed time‐between‐event observations. The first two methods are the Weibull CUSUM chart and the exponential CUSUM (ECUSUM) chart. The latter is considered in literature to be robust to the assumption of the exponential distribution when observations have a Weibull distribution. For the third CUSUM chart included in this study, an adjustment in the design of the ECUSUM chart is used to account for the true underlying time‐between‐event distribution. This adjustment allows for the adjusted ECUSUM chart to be directly comparable to the Weibull CUSUM chart. By comparing the zero‐state average run length and average time to signal performance of the three charts, the ECUSUM chart is shown to be much less robust to departures from the exponential distribution than was previously claimed in the literature. We demonstrate the advantages of using one of the other two charts, which show surprisingly similar performance. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

A new variable control chart under failure‐censored reliability tests for Weibull distribution

In this paper, we propose a new variable control chart under type II or failure‐censored reliability tests by assuming that the lifetime of a part follows the Weibull distribution with fixed and stable shape parameter. The purpose is to monitor the mean and the variance of a Weibull process. In fact, the mean and the variance are related to the scale parameter. The necessary measures are given to calculate the average run length (ARL) for in‐control and shifted processes. The tables of ARLs are presented for various shift constants and specified parameters. A simulation study is given to show the performance of the proposed control chart. The efficiency of the proposed control chart is compared with a control chart based on the conditional expected value under type II censoring. An example is also given for the illustration purpose.     

Reliability analysis for electronic devices using beta‐Weibull distribution

Today, in reliability analysis, the most used distribution to describe the behavior of electronic products under voltage profiles is the Weibull distribution. Nevertheless, the Weibull distribution does not provide a good fit to lifetime datasets that exhibit bathtub‐shaped or upside‐down bathtub–shaped (unimodal) failure rates, which are often encountered in the reliability analysis of electronic devices. In this paper, a reliability model based on the beta‐Weibull distribution and the inverse power law is proposed. This new model provides a better approach to model the performance and fit of the lifetimes of electronic devices. To estimate the parameters of the proposed model, a Bayesian analysis is used. A case study based on the lifetime of a surface mounted electrolytic capacitor is presented, the results showed that the estimation of the proposed model differs from the inverse power law–Weibull and that it affects directly the mean time to failure, the failure rate, the behavior, and the performance of the capacitor under analysis.     

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.     

Analysis of field failure data when modeled Weibull slope increases with time

Weibull time‐to‐fail distributions cannot be correctly estimated from field data when manufacturing populations from different vintages have different failure modes. To investigate the pitfalls of ongoing Weibull parameter estimation, two cases, based upon real events, were analyzed. First, a time‐to‐fail distribution was generated assuming the same Weibull shape parameter representing an increasing failure rate for each monthly batch or vintage of production. The shape parameter was estimated from simulated field data at regular periods as the population accumulated service time. Estimates of the shape parameter were not constant, but gradually decreased (as had occurred in a real system) with added service time. In the second case, field reliability performance was modeled to match the actual historical data for one product from a disk drive manufacturer. The actual data was proprietary and was not directly available for analysis. A production schedule was modeled with a mix of two failure characteristics. The population reaching the field in the first 12 months had a low, constant failure rate. For the second and third years of production, higher volumes were introduced that had the higher, increasing failure rates of the first case. Assessment of the mixed population at each month of calendar time resulted in an increasing Weibull shape parameter estimate at each assessment. When the two populations were separated and estimated properly, a better fit with more accurate estimates of Weibull shape parameters resulted. Copyright © 2010 John Wiley & Sons, Ltd.     

An exponentially weighted moving average chart based on likelihood‐ratio test for monitoring Weibull mean and variance with subgroups

Monitoring changes in the Weibull mean and variance simultaneously is of interest in quality control. The mean and variance of a Weibull process are determined by its shape and scale parameters. Most studies are focused on monitoring the Weibull scale parameter with fixed shape parameter or the Weibull shape parameter with fixed scale parameter. In this paper, we propose an exponentially weighted moving average chart based on the likelihood‐ratio test and an inverse error function called ELR chart to monitor changes in the Weibull mean and variance simultaneously. The simulation approach is used to derive the average run length. We compare our proposed chart with other existing control charts for 3 cases, including scale parameter changes with fixed shape parameter, shape parameter changes with fixed scale parameter, and both parameters changes. The results show that the ELR chart outperforms the other control charts in terms of average run length in most cases. Two numerical examples are used to illustrate the applications of the proposed control chart.     

Design and comparison of some Shewhart‐type schemes for simultaneous monitoring of Weibull parameters

Weibull distribution is one of the most important probability models used in modeling time between events, system reliability, and particle sizes, among others. Therefore, efficiently and consequently monitoring certain changes in Weibull process is considered as an important research topic. Various statistical process monitoring schemes have been developed for monitoring different process parameters, including some for Weibull parameters. Most of these schemes are, however, designed to monitor and control a single process parameter, although there are two important model parameters for Weibull distribution. Recently, several researchers studied various schemes for jointly monitoring the mean and variance of a normally distributed process using a single plotting statistic. Nevertheless, there is still dearth of researches in joint monitoring of non‐normal process parameters. In this context, we develop some control schemes for simultaneously monitoring the scale and shape parameters of processes that follow the Weibull distribution. Implementation procedures are developed, and performance properties of various proposed schemes are investigated. We also offer an illustrative example along with a summary and recommendations.     

Multi-Censored Sampling in the Three Parameter Weibull Distribution

In life and fatique testing, multi-censored samples arise when at various stages of a test, some of the survivors are withdrawn from further observation. Sample specimens which remain after each stage of censoring continue to be observed until failure or until a subsequent stage of censoring. In this paper, maximum likelihood estimators and estimators which utilize the first order statistic are derived for the three parameter Weibull distribution. Estimators are also derived for the special case in which the shape parameter is known, a special case which includes the two parameter exponential distribution. An illustrative example is included.     

Monitoring the Shape Parameter of a Weibull Regression Model in Phase II Processes

In this paper, the interest is focused on monitoring profiles with Weibull distributed‐response and common shape parameter γ in phase II processes. The monitoring of such profiles is completely possible by taking the natural logarithm of the Weibull‐distributed response. This is equivalent to characterize the correspondent process by an extreme value linear regression model with common scale parameter σ = γ^?1. It was found out that from the monitoring of the common log‐scale parameter of the extreme value linear regression model, with the help of a simple scheme, it can be obtained important information about the deterioration of the entire process assuming the β coefficients as nuissance parameters that do not have to be known but stable. Control charts are based on the relative log‐likelihood ratio statistic defined for the log‐scale parameter of the log‐transformation of the Weibull‐distributed response and its respective signed square root. It was also found out that some existing adjustments are needed in order to improve the accuracy of using the distributional properties of the monitoring statistics for relatively small and moderate sample sizes. Simulation studies suggest that resulting charts have appealing properties and work fairly acceptable when non‐large enough samples are available at discrete sampling moments. Detection abilities of the studied corrected control schemes improve when sample size increases. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Reliability Analysis of Zero‐Failure Data with Poor Information

For costly and dangerous experiments, growing attention has been paid to the problem of the reliability analysis of zero‐failure data, with many new findings in world countries, especially in China. The existing reliability theory relies on the known lifetime distribution, such as the Weibull distribution and the gamma distribution. Thus, it is ineffective if the lifetime probability distribution is unknown. For this end, this article proposes the grey bootstrap method in the information poor theory for the reliability analysis of zero‐failure data under the condition of a known or unknown probability distribution of lifetime. The grey bootstrap method is able to generate many simulated zero‐failure data with the help of few zero‐failure data and to estimate the lifetime probability distribution by means of an empirical failure probability function defined in this article. The experimental investigation presents that the grey bootstrap method is effective in the reliability analysis only with the few zero‐failure data and without any prior information of the lifetime probability distribution. Copyright © 2011 John Wiley & Sons, Ltd.     

Design of Over-Stress Life-Test Experiments When Failure Times Have the Two-Parameter Weibull Distribution

This paper deals with estimation of reliability parameters when life testing is conducted at stress levels above that which would normally be applied in standard usage. A range of stress for testing is prescribed, and a two-parameter Weibull model for failure times is assumed. The logarithm of the Weibull scale parameter is assumed to be a polynomial function of known degree k of the reciprocal of stress level. The Weibull shape parameter is assumed to be independent of stress level.

The problem considered is that of determining the design for obtaining the least-squares-curve intercept with minimum variance at the nominal testing level. The design obtained specifies the number and location of stress levels in the prescribed range at which the life tests will be conducted and proportion of the total sample of specified size to be randomly allocated to each testing level.     

The joint economic‐statistical design of X and R charts for nonnormal data

We consider the joint economic‐statistical design of X and R control charts under the assumption that the quality measurement and the in‐control time have Johnson and Weibull distributions. The Johnson distribution is general in that it can be made to fit all possible values of skewness and kurtosis. The four parameters—the sample size n, time h between successive samples, and the control factors k₁ and k₂ for the X and R charts—are determined so that the mean hourly loss‐cost is minimized under constraints on the Type I and II error probabilities. We have generalized the Costa model to accommodate the Johnson and Weibull distributions. Sensitivity to nonnormality, shift, and Weibull scale parameter is considered in our analysis. Our sensitivity analysis shows that the optimal design parameters are sensitive to nonnormality. Comparisons of the fully economic and economic‐statistical designs are given. Copyright © 2010 John Wiley & Sons, Ltd.     

Control Charts For Monitoring The Weibull Shape Parameter Based On Type‐II Censored Sample

In this paper, a new statistic is proposed to monitor the Weibull shape parameter when the sample is type II censored. The one‐sided and two‐sided average run length‐unbiased control charts are derived based on the new monitoring statistic. The control limits of the proposed control charts depend on the sample size, the failure number and the false alarm rate. Using Monte Carlo simulation, the performance of the proposed control charts is studied and compared with the range‐based charts proposed by Pascual and Li (2012), which is equivalent to the proposed control charts when r = 2. The simulation results show that the proposed control charts perform better than the ones of Pascual and Li (2012). This paper also evaluates the effects of parameter estimation on the proposed control charts. Finally, an example is used to illustrate the proposed control charts. Copyright © 2012 John Wiley & Sons, Ltd.     

On misinterpretation of the asymptotic property of system time‐between‐failures

If a system, not necessarily one in series, is composed of components having various time‐to‐failure distributions and components are replaced good‐as‐new as they fail, then the system time‐between‐failure distribution tends toward the exponential. Many practicing reliability engineers, incorrectly invoking this property, model their systems with an exponential time‐to‐failure. We show, under two conditions, using a hypothetical fleet of vehicles, the severity of this error. Modeling time‐to‐failure as exponential results in gross over‐sparing and high unavailability costs as well. Copyright © 2001 John Wiley & Sons, Ltd.     

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.