Exploring generalized probability weighted moments, generalized moments and maximum likelihood estimating methods in two-parameter Weibull model

Generalized probability weighted moments (GPWM), generalized moments and maximum likelihood (ML) estimating methods are investigated in the two-parameter Weibull (WEI) model. Point estimators for positive and negative shape parameters and for quantiles with special return periods are derived. Analytical expressions for the asymptotic variances of the estimators are presented. Simulation results on the performance of the three estimating methods are also given. The results show that the GPWM method may in some situations lead to a slight gain in quantile estimation accuracy. However, the overall results show the ML method to be the most recommendable one, since for the cases considered it performed either better or almost as good as the GPWM method. The WEI model is then used to fit a hydrological data set of flood volumes above a threshold, using data from the Little Southwest Miramichi River, in New Brunswick, Canada. It is shown that one added advantage for using ML method to fit the two-parameter WEI model is that small-sample procedures are available for calculating confidence intervals for WEI quantiles, when this method is used. It is recommended that such small-sample methods be used whenever available, in hydrology, for estimating distribution quantiles.