An Addendum to “Linear Estimation of the Weibull Parameters”

Linear, asymptotically normal and efficient estimators are given for the shape parameter of the two parameter Weibull distribution when the scale parameter is known and for the log of the scale parameter when the shape parameter is known. The weights of the ordered observations and other constants needed for these estimators are readily obtainable from a previous article of the author.     

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.     

Robust Estimation for Weibull Distribution in Partially Accelerated Life Tests with Early Failures

Maximum likelihood estimation (MLE) is a frequently used method for estimating distribution parameters in constant stress partially accelerated life tests (CS‐PALTs). However, using the MLE to estimate the parameters for a Weibull distribution may be problematic in CS‐PALTs. First, the equation for the shape parameter estimator derived from the log‐likelihood function is difficult to solve for the occurrence of nonlinear equations. Second, the sample size is typically not large in life tests. The MLE, a typical large‐sample inference method, may be unsuitable. Test items unsuitable for stress conditions may become early failures, which have extremely short lifetimes. The early failures may cause parameter estimate bias. For addressing early failures in the Weibull distribution in CS‐PALTs, we propose an M‐estimation method based on a Weibull Probability Plot (WPP) framework, which leads a closed‐form expression for the shape parameter estimator. We conducted a simulation study to compare the M‐estimation method with the MLE method. The results show that, with early‐failure samples, the M‐estimation method performs better than the MLE does. Copyright © 2015 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.     

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.     

Use of median-based estimator to construct Phase II exponential chart

The Shewhart-type exponential control chart is a popular and extensively used among all time-between-events control charts for its simplicity. When the parameter is unknown, Phase II control limits are constructed, and the success of its implementation depends to an extent on the estimated value of the parameter, obtained from Phase I dataset. However, when the Phase I data are contaminated with spurious observations/outliers, the performance of the chart is suspected to deviate from what is normally expected. Traditionally, maximum likelihood estimator (MLE) and minimum variance unbiased estimator (MVUE) are used to estimate the unknown process parameter. Both of estimators are the functions of sample mean. In this paper, the median-based estimator (MBE) that is a function of sample median is used to construct Phase II control limits. Moreover, performance of the proposed chart is examined when Phase I sample consists of contaminated observations/outliers. It is found that the proposed chart outperforms the existing charts whether the sample is contaminated or not.     

On estimating percentiles of the Weibull distribution by the linear regression method

Probability estimators developed previously by the authors have been used to obtain unbiased estimates of the Weibull parameters by the linear regression method. Using these unbiased estimators, percentiles of the Weibull distribution have been estimated. Since these percentiles are determined from the estimated parameters, they also have distributions and subsequently are determined for five sample sizes. Analysis has shown that the distributions of these estimated percentiles are neither normal, lognormal, three-parameter Weibull nor three-parameter log-Weibull. A new methodology to estimate the percentile with a specified level of confidence has been introduced. The step-by-step use of the methodology is demonstrated by examples in this paper.     

Pseudo maximum likelihood estimates using ranked set sampling with applications to estimating correlation

Suppose θ is the parameter of interest and λ is the nuisance parameter. When obtaining the maximum likelihood estimator (MLE) of θ in the presence of λ requires intensive computation, the pseudo MLE of θ, based on a pseudo likelihood function, can be used. Gong and Samaniego (Ann. Stat. 9:861–869, 1981) proposed a pseudo MLE (PMLE) based on simple random samples. Ranked set sampling has been applied to the bivariate variables (X,Y) where measuring one of the variables is difficult or costly. In this paper, we obtain the pseudo MLE of the correlation coefficient from a bivariate normal distribution (X,Y) based on ranked set samples, assuming that Y is difficult or more expensive to measure and that the mean and variance of Y are the nuisance parameters. The PMLE is compared with three other estimators of the correlation coefficient. Simulations show that the PMLE is more (less) efficient than other estimators, depending on value of ρ. Testing of soil contamination provides an example of the use of the methods.     

GLM profile monitoring using robust estimators

It is well known that performance of control scheme in phase II of statistical process control depends on the estimators utilized in phase I. Sometimes, outliers may be present in the data, which could seriously impact the performance of the estimators. In some practical situations, generalized linear models (GLMs) are used to model a wide class of response variables. This study deals with the robust estimation and monitoring of parameters in GLM profiles in the presence of outliers. In this study, robust estimators are used to estimate the parameters of logistic and Poisson profiles. The results are compared with the maximum likelihood estimators (MLEs). In a numerical example, the profile parameters are estimated by the MLE and robust estimators and the resulting test statistics are monitored by a control scheme. The phase II control charts are determined based on these two types of estimators and compared for different out-of-control conditions. The simulation results confirm that robust estimators in most cases lead to better estimates in comparison with the MLE estimator in terms of average run length criterion.     

Unbiased estimates of the Weibull parameters by the linear regression method

For sample sizes from 5 to 100, the bias of the scale parameter was investigated for probability estimators, P = (i − a)/(n + b), which yield unbiased estimates of the shape parameter. A class of unbiased estimators for both the shape and scale parameters was developed for each sample size. In addition, the percentage points of the distribution of unbiased estimate of the shape parameter were determined for all sample sizes. The distribution of the scale parameter was found to be normal by using the Anderson-Darling goodness-of-fit test. How the results can be used to establish confidence intervals on both the shape and scale parameters are demonstrated in the paper.     

The problem of the selection of an a posteriori error indicator based on smoothening techniques

This paper addresses the problem of assessing the quality of an a posteriori error estimate of a finite element solution. An error estimate based on local L2-projections is analysed in the case of translation-invariant meshes. It is shown that for general meshes this technique does not lead to an asymptotically exact estimator. The problem is analysed in detail in the one-dimensional setting. It is shown that an asymptotically exact estimator is not the optimal one when the solution is not sufficiently smooth. An optimal estimator for adaptively constructed meshes is given. Finally, a general mathematical framework for the quality assessment of estimators is introduced.     

Inference with progressively censored <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system lifetime data

A system with n independent components which works if and only if a least k of its n components work is called a k-out-of-n system. For exponentially distributed component lifetimes, we obtain point and interval estimators for the scale parameter of the component lifetime distribution of a k-out-of-n system when the system failure time is observed only. In particular, we prove that the maximum likelihood estimator (MLE) of the scale parameter based on progressively Type-II censored system lifetimes is unique. Further, we propose a fixed-point iteration procedure to compute the MLE for k-out-of-n systems data. In addition, we illustrate that the Newton–Raphson method does not converge for any initial value. Finally, exact confidence intervals for the scale parameter are constructed based on progressively Type-II censored system lifetimes.     

Goodness-of-fit tests for symmetric stable distributions—Empirical characteristic function approach

We consider goodness-of-fit tests for symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponent α estimated from the data. We treat α as an unknown parameter, but for theoretical simplicity we also consider the case that α is fixed. For estimation of parameters and the standardization of data we use the maximum likelihood estimator (MLE). We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distribution of the test statistic is obtained using complex integration. We find that if the sample size is large the calculated asymptotic critical values of test statistics coincide with the simulated finite sample critical values. Finite sample power of the proposed test is examined. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.     

Maximum likelihood blood velocity estimator incorporating properties of flow physics

The aspect of correlation among the blood velocities in time and space has not received much attention in previous blood velocity estimators. The theory of fluid mechanics predicts this property of the blood flow. Additionally, most estimators based on a cross-correlation analysis are limited on the maximum velocity detectable. This is due to the occurrence of multiple peaks in the cross-correlation function. In this study a new estimator (CMLE), which is based on correlation (C) properties inherited from fluid flow and maximum likelihood estimation (MLE), is derived and evaluated on a set of simulated and in vivo data from the carotid artery. The estimator is meant for two-dimensional (2-D) color flow imaging. The resulting mathematical relation for the estimator consists of two terms. The first term performs a cross-correlation analysis on the signal segment in the radio frequency (RF)-data under investigation. The flow physic properties are exploited in the second term, as the range of velocity values investigated in the cross-correlation analysis are compared to the velocity estimates in the temporal and spatial neighborhood of the signal segment under investigation. The new estimator has been compared to the cross-correlation (CC) estimator and the previously developed maximum likelihood estimator (MLE). The results show that the CMLE can handle a larger velocity search range and is capable of estimating even low velocity levels from tissue motion. The CC and the MLE produce incorrect velocity estimates due to the multiple peaks, when the velocity search range is increased above the maximum detectable velocity. The root-mean square error (RMS) on the velocity estimates for the simulated data is on the order of 7 cm/s (14%) for the CMLE, and it is comparable to the RMS for the CC and the MLE. When the velocity search range is set to twice the limit of the CC and the MLE, the number of incorrect velocity estimates are 0, 19.1, and 7.2% for the CMLE, CC, and MLE, respectively. The ability to handle a larger search range and estimating low velocity levels was confirmed on in vivo data.     

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.     

Asymptotic Properties of Several Estimators of Weibull Parameters

In this paper several estimators for the scale and the shape parameters of the Weibull law are obtained by postulating a stochastic model aimed at studying the aging process of certain inexpensive industrial products. These estimators are consistent and asymptotically multi-normal.     

Semi-parametric second-order reduced-bias high quantile estimation

In many areas of application, like, for instance, Climatology, Hydrology, Insurance, Finance, and Statistical Quality Control, a typical requirement is to estimate a high quantile of probability 1−p, a value high enough so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index γ, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for γ>0, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles’ estimators, with the same asymptotic variance as the (biased) “classical” estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.     

Estimating the Expected Probability of Misclassification for a Rule Based on the Linear Discriminant Function: Univariate Normal Case

The problem is to estimate the average probability of misclassifying an observation from a given population in the context of the two group classification problem when populations are univariate normal with unknown means and common known variance, and the rule is based on the linear discriminant function. Several estimators are compared with respect to asymptotic MSE and with respect to the distribution of the absolute error between estimator and parameter, and conclusions drawn about the best estimators.     

Estimation of Weibull parameters with linear regression method

Abstract

Based on minimum mean square error, a modified probability estimator is proposed by a Monte Carlo simulation for estimating the Weibull parameters with the linear regression method. It is shown that compared with the commonly used estimators, the modified probability estimator gives a more accurate estimation of the Weibull modulus and the same estimation precision of the scale parameter. Furthermore, it is more conservative than the commonly used estimator recommended by previous authors and hence results in a higher safety in reliability predictions.