Maximum likelihood vs. maximum goodness of fit estimation of the three-parameter Weibull distribution

Abstract

The use of statistics based on the empirical distribution function is analysed for estimation of the scale, shape, and location parameters of the three-parameter Weibull distribution. The resulting maximum goodness of fit (MGF) estimators are compared with their maximum likelihood counterparts. In addition to the Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling statistics, some related empirical distribution function statistics using different weight functions are considered. The results show that the MGF estimators of the scale and shape parameters are usually more efficient than the maximum likelihood estimators when the shape parameter is smaller than 2, particularly if the sample size is large.

Keywords:

• Anderson–Darling
• Cramer–von Mises
• Empirical distribution function
• Kolmogorov distance
• Minimum distance estimators

Acknowledgements

This research was partially supported by the Spanish DGI Grant MTM2005-00287. I thank Professor George E. P. Box for the great help during my repeated visits to the University of Wisconsin–Madison, where part of the research was performed. I am also grateful to an associate editor and a referee for their very helpful suggestions, which have led to an improved manuscript.