Fitting the log-logistic distribution by generalized moments

SummaryThe method of generalized moments (GM) is investigated for parameter and quantile estimation in the 2-parameter log-logistic (LL2) model. Point estimators for the shape and scale parameters and quantiles are derived. Asymptotic variances and covariances for these estimators are presented, along with simulation results on the performance of the GM method versus the methods of generalized probability weighted moments (GPWM), of maximum likelihood (ML), and of classical moments applied to Y = ln X. The GPWM and ML methods have already been investigated by the authors. Some mathematical properties of the LL2 model and some relationships between GM and GPWM are highlighted. The simulation results show the GM method to outperform the other competitive methods in the LL2 case, when moment orders are appropriately chosen. It is also shown that a mixture of moments of positive and negative orders is needed for optimal estimation under an LL2 model, and how this mixture can be implemented using the GM method. However, further research into the area of optimal choice of moment orders is still needed. Mixing positive and negative moments in the estimation is demonstrated by a hydrological example involving low stream flow.