Parameter and quantile estimation for the generalized extreme-value distribution: a second look

Abstract

The Generalized Extreme Value distribution (GEVD) was introduced by Jenkinson (1955; Quarterly Journal of the Meteorological Society81, 158–171) as a model for extremes of natural phenomena such as sea level, rainfall, river heights and air pollutants. The GEVD is a three-parameter family with a location, scale, and shape parameter. Given a set of data, there were two traditional methods for estimating the parameters: maximum likelihood and probability-weighted moments (PWM), the latter being better (Hosking et al., 1985; Technometrics27, 251–261). Castillo and Hadi (1994; Environmetrics5, 417–432) then introduced two methods based on order statistics, LMS and MED, as improvements over PWM, claiming that the performance of the PWM worsens as the shape parameter increased. Unfortunately, their comparison of the methods was flawed as they used an inappropriate approximation to the PWM equations. Thus, PWM were inaccurately discarded. In this paper, PWM are reexamined and the correct computational procedures are discussed. We show that, even for large shape parameter, PWM still compare well to estimators based on order statistics. In fact, simulation results show that PWM estimators yield smaller bias and root mean square error (RMSE) than LMS for the estimation of the parameters and the quantiles. Furthermore, PWM yield smaller bias and RMSE than MED for the estimation of the parameters, and only slightly larger bias and RMSE for the quantiles. Based on bias and RMSE, and given the existence of asymptotic results, PWM are still a better choice. Copyright © 1999 John Wiley & Sons, Ltd.