Concerns About Maximum Likelihood Estimation for the Three-Parameter Weibull Distribution: Case Study of Statistical Software

Abstract

Maximum likelihood estimation of the two-parameter Weibull distribution is straightforward; however, there are multiple methods for maximum likelihood estimation of the three-parameter Weibull. The third parameter for the three-parameter Weibull distribution shifts the origin from 0 to some generally positive value sometimes called the location, threshold, or minimum life. This article initially evaluates twelve statistical packages for four real-world datasets including oil spill data from the Gulf of Mexico. The different methods used by the packages result in fairly major differences in the estimated parameters between the statistical packages. Some statistical packages do not offer three-parameter Weibull estimation. Other software attempts the estimation in only certain cases. This may have major implications for those needing to estimate or apply the results of a three-parameter Weibull distribution that is used frequently in practice. A subset of the twelve packages is subsequently analyzed in detail based on an experimental design using pseudo-random Weibull datasets. This article also discusses the most common estimation method employed, which is maximizing a profile log-likelihood function.