A review of maximum likelihood estimation methods for the three-parameter weibull distribution

The three-parameter Weibull distribution has found extensive applications in various fields of study. Among many parameter estimation methods available the method of maximum likelihood (ML) is considered to be one of the most reliable. ML estimators are known to have desirable properties of consistency, asymptotic normality and asymptotic efficiency for large samples under fairly general conditions. An up-to-date review of the currently available ML estimation methods for the three-parameter Weibull is presented here. Most of these methods have been developed for the complete samples, while only a few have been applied to the more difficult cases of censored or grouped samples. The comparison of these methods for the complete sample case produces no clearly superior method (in accuracy and speed). The computational approaches developed for censored or grouped samples are based on the methods for the complete samples and exhibit similar characteristics. Many of the computational difficulties can be traced to the theoretical complexity of the underlying nonlinear optimization problem for ML estimation. Finally, several directions of future research are suggested, among them the desirability to develop new numerical methods more closely tailored to the special structure of the underlying optimization problems. There is also a need for a standard software which could be used by all those interested in Weibull ML estimation.