Fitting the Three-Parameter Weibull Distribution: Review and Evaluation of Existing and New Methods

The three-parameter Weibull distribution is a commonly-used distribution for the study of reliability and breakage data. However, given a data set, it is difficult to estimate the parameters of the distribution and that, for many reasons: (1) the equations of the maximum likelihood estimators are not all available in closed form. These equations can be estimated using iterative methods. However, (2) they return biased estimators and the exact amount of bias is not known. (3) The Weibull distribution does not meet the regularity conditions so that in addition to being biased, the maximum likelihood estimators may also be highly variable from one sample to another (weak efficiency). The methods to estimate parameters of a distribution can be divided into three classes: a) the maximizing approaches, such as the maximum likelihood method, possibly followed by a bias-correction operation; b) the methods of moments; and c) a mixture of the previous two classes of methods. We found using Monte Carlo simulations that a mixed method was the most accurate to estimate the parameters of the Weibull distribution across many shapes and sample sizes, followed by the weighted Maximum Likelihood estimation method. If the shape parameter is known to be larger than 1, the Maximum Product of Spacing method is the most accurate whereas in the opposite case, the mixed method is to be preferred. A test that can detect if the shape parameter is smaller than 1 is discussed and evaluated. Overall, the maximum likelihood estimation method was the worst, with errors of estimation almost twice as large as those of the best methods.