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

Abstract

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

Analysis of the fatigue life estimators of the materials using small samples

“…When a = 3.48, it becomes a distribution close to normal. 10,28 The estimation of parameters in a model is one of the main steps in the analysis of a distribution, and it is known that a good estimator must be ''close'' in some way to the value of the true unknown parameter, that is, the estimator must be a function of sufficient statistics. 31 The four methods of estimation will be approached in this article: MLE, MOM, LSE and WLSE.…”

“…. , b 10, 000 , through which we obtain the (E), bias and root mean square error (RMSE) equations (23)- (28) Eb À Á = 1 10, 000…”

“…The work of Hung, 26 proposed to use the WLSE to estimate the shape parameter of the Weibull distribution and its results, by simulation, presented better performance than the techniques already mentioned. Some publications [27][28][29] compared and analyzed WLSE performance, but nothing applied with real data and small samples. In the study by Zhang et al, 9 a WLSE model with adjusted weights that considerably improves the performance from the estimate to a small set of data was proposed.…”

Process Capability Indices for Shape Parameter of Weibull Distribution

“…The function also observed relationship between the shape of the fatigue life distribution and the stress level. Zhang, Xie and Tang [24] used Weibull shape parameter as a measure of reliability and compared various estimators based on different assumptions. The Weibull shape parameter has significance to determine the behavior of the failure rate of the product or system in measuring time to failure in any electrical or mechanical system.…”

Comparing performances of clements, box-cox, Johnson methods with weibull distributions for assessing process capability

“…Weibull distribution covers a wide class of non-normal processes due to its capability to yield a variety of distinct curves based on its parameters. The shape parameter of Weibull distribution determines the behavior of the failure rate of the product or system and has been used as a measure of reliability (Yavuz, 2013). Hsu, Pearn and Lu (2011) use Weibull distributions to model the data of the processes and express time until a given technical device fails.…”