Alternative PWM-estimators of the Gumbel distribution

Probability weighted moments (PWM) are widely used in hydrology for estimating parameters of statistical distributions, including the Gumbel distribution. The classical PWM-approach considers the moments β_i=E[XFⁱ] with i=0,1 for estimation of the Gumbel scale and location parameters. However, there is no reason why these probability weights (F⁰ and F¹) should provide the most efficient PWM-estimators of Gumbel parameters and quantiles. We explore an extended class of PWMs that does not impose arbitrary restrictions on the values of i. Estimation based on the extended class of PWMs is called the generalized method of probability weighted moments (GPWM) to distinguish it from the classical procedure. In fact, our investigation demonstrates that it may be advantage to use weight functions that are not of the form Fⁱ. We propose an alternative PWM-estimator of the Gumbel distribution that maintains the computational simplicity of the classical PWM method, but provides slightly more accurate quantile estimates in terms of mean square error of estimation. A simple empirical formula for the standard error of the proposed quantile estimator is presented.