A study of two estimation approaches for parameters of Weibull distribution based on WPP

Abstract

Least-squares estimation (LSE) based on Weibull probability plot (WPP) is the most basic method for estimating the Weibull parameters. The common procedure of this method is using the least-squares regression of Y on X, i.e. minimizing the sum of squares of the vertical residuals, to fit a straight line to the data points on WPP and then calculate the LS estimators. This method is known to be biased. In the existing literature the least-squares regression of X on Y, i.e. minimizing the sum of squares of the horizontal residuals, has been used by the Weibull researchers. This motivated us to carry out this comparison between the estimators of the two LS regression methods using intensive Monte Carlo simulations. Both complete and censored data are examined. Surprisingly, the result shows that LS Y on X performs better for small, complete samples, while the LS X on Y performs better in other cases in view of bias of the estimators. The two methods are also compared in terms of other model statistics. In general, when the shape parameter is less than one, LS Y on X provides a better model; otherwise, LS X on Y tends to be better.

Introduction

The Weibull distribution is widely used in life testing and reliability studies. Some modifications have been developed in recent years, which greatly enlarged the applications. For some recent references, see e.g. [1], [2], [3], [4].

The cumulative distribution function (CDF) of the two-parameter Weibull distribution is

$$F(t;\alpha ,\beta )=1-\exp \left[-{\left(\frac{t}{\alpha }\right)}^{\beta }\right]\text{,}$$

where t is the time to failure, α is the scale parameter and β is the shape parameter. The shape parameter characterizes the failure rate trend [5] and thus can indicate some failure modes (initial, random and wear-out) [6].

Many methods have been proposed for estimating α and β, such as the method of moments and the maximum likelihood estimation (MLE). Robust estimation methods and bias correction methods are of recent interests; see, e.g., [7], [8]. For simplicity, a common practice is to plot the failure time t vs. the estimated F(t) on a Weibull probability paper, and then fit a straight line to the data points. β can be estimated by the slope of the straight line and α by the intercept and slope. This method is preferred by engineers because of its computation simplicity and the graphical presentation [9]. The Weibull probability plot (WPP), in addition to providing simple straight line fitting for parameter estimation, serves other purposes, including evaluating whether the data are Weibull distributed and identifying outliers [10].

After data points are placed on the Weibull paper, the least-squares (LS) method is usually used to fit a straight line instead of by eye. The straight line can be fit by minimizing the sum of squares of the vertical residuals, which is called LS Y on X method, or it can be fit by minimizing the sum of squares of the horizontal residuals, which is called LS X on Y method. This paper compares these two methods for estimating the Weibull parameters in cases of both complete and censored data. The motivation of our study is that we noticed LS X on Y was adopted in the early literature, see e.g., [11], [12], [13], but is replaced by LS Y on X in the recent literature with a few exceptions, see, e.g., [14], [15].

To the best of our knowledge, there is no intensive comparison of the two methods existing in the literature, especially for censored data. In this paper, the two methods are compared in terms of both properties of the parameter estimators and model statistics of the two linear regression models. Both complete and multiply censored samples are examined. Suggestions of when to use which method are given based on the study.