A Note on Weighted Least‐squares Estimation of the Shape Parameter of the Weibull Distribution

Lu, Hai‐Lin; Chen, Chong-Hong; Wu, Jong-Wuu

doi:10.1002/qre.570

Cited by 29 publications

(31 citation statements)

References 11 publications

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“…On the other hand, it should be noted that the error terms of the models given in Equations (11) and (12) are not identically distributed as mentioned by [9,[15][16][17]19,20], since x piq is beta distributed with parameters i and n´i`1 and, thus, the models (11) and (12) have a different variance for each i as earlier mentioned [9,19,20]. In other words, error terms of the previously mentioned models have no equal variance.…”

Section: Lse Methods For Log-logistic and Weibull Distributionsmentioning

confidence: 99%

“…While, according to the MSE criterion, the proposed WLSE, LSE, and WLSE (Zyl and Schall) yield similar results, the proposed WLSE provides comparable efficiency in comparison with LSE in terms of bias. Moreover, the proposed WLSE is compared to two alternative WLSE (Hung [11], Lu et al [12]) and also PE [2] for the shape parameter of the Weibull distribution. It can be seen that the proposed WLSE demonstrates satisfactory results in terms of MSE.…”

Section: Monte Carlo Simulationmentioning

confidence: 99%

“…However, heteroscedasticity (non-constant variance) is present in the used regression model and, thus, LS estimates lose the efficiency property, even if most studies do not take into account this reality [9]. In such cases, the WLSE or alternative methods should be used [10][11][12][13][14][15][16][17][18][19][20]. It is well-known that when conducting the WLSE, the variances of the dependent variables are unknown and must be estimated to implement the WLSE procedure.…”

Section: Introductionmentioning

confidence: 99%

“…It is well-known that when conducting the WLSE, the variances of the dependent variables are unknown and must be estimated to implement the WLSE procedure. Considering this reality, [11][12][13][14][15][16] discuss that variances should be found in order to perform the WLSE. They all generally consider different weights using large sample properties of the empirical distribution function or order statistics to stabilize the variance when conducting the WLSE.…”

Section: Introductionmentioning

confidence: 99%

“…His results from simulation studies illustrate that the mean-squared error of WLSE is smaller than competing procedures. Lu et al [12] consider the LSE and WLSE for the Weibull distribution and compare their WLSE method with three existing WLSE methods. They found that their WLSE method is more precise and has smaller variance than Bergman's WLS estimators.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Estimating Variances in Weighted Least-Squares Estimation of Distributional Parameters

Kantar

2016

MCA

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Many estimation methods have been proposed for the parameters of statistical distribution. The least squares estimation method, based on a regression model or probability plot, is frequently used by practitioners since its implementation procedure is extremely simple in complete and censoring data cases. However, in the procedure, heteroscedasticity is present in the used regression model and, thus, the weighted least squares estimation or alternative methods should be used. This study proposes an alternative method for the estimation of variance, based on a dependent variable generated via simulation, in order to estimate distributional parameters using the weighted least squares method. In the estimation procedure, the variances or weights are expressed as a function of the rank of the data point in the sample. The considered weighted estimation method is evaluated for the shape parameter of the log-logistic and Weibull distributions via a simulation study. It is found that the considered weighted estimation method shows better performance than the maximum likelihood, least-squares, and certain other alternative estimation approaches in terms of mean square error for most of the considered sample sizes. In addition, a real-life example from hydrology is provided to demonstrate the performance of the considered method.

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Section: Lse Methods For Log-logistic and Weibull Distributionsmentioning

confidence: 99%

Section: Monte Carlo Simulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Estimating Variances in Weighted Least-Squares Estimation of Distributional Parameters

Kantar

2016

MCA

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

Yavuz

2012

Quality & Reliability Eng

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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.where b is the scale parameter and c is the shape parameter. When c = 1, the Weibull distribution reduces to the exponential distribution, and when c = 2, the Weibull distribution is Rayleigh. When c = 3.48, the distribution is close to Normal.The Weibull shape parameter (c) is very important because it determines the behavior of the failure rate of the product or system, 1 and has been used as a measure of reliability. It can be estimated from a sample using either linear regression, maximum likelihood, or the method of moments. The problem of estimating the Weibull shape parameter is not new; it has been studied for more than 30 years. However, some open issues still remain, especially for the linear regression methods. 2 Regression analysis is a simple method for investigating and modeling the functional relationship between variables. The simplest case is the so-called simple linear regression model, which contains only one independent variable and, can be written asThe WLS method was suggested based on the fact that some observation values used in regression analysis are quite different from others. This method aims to eliminate the adverse effects of contradictory values on the model by placing a different weight on each A. A. YAVUZ

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

Cheng

Sheu

2015

Quality & Reliability Eng

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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.

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A Note on Weighted Least‐squares Estimation of the Shape Parameter of the Weibull Distribution

Cited by 29 publications

References 11 publications

Estimating Variances in Weighted Least-Squares Estimation of Distributional Parameters

Estimating Variances in Weighted Least-Squares Estimation of Distributional Parameters

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

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

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