Parameter and quantile estimation for the generalized extreme‐value distribution

Abstract

AbstractThe generalized extreme‐value distribution (GEVD) was introduced by Jenkinson (1955). It is now widely used to model extremes of natural and environmental data. The GEVD has three parameters: a location parameter (−∞ < Λ < ∞), a scale parameter (α > 0) and a shape parameter (−∞ < k < ∞). The traditional methods of estimation (e.g., the maximum likelihood and the moments‐based methods) have problems either because the range of the distribution depends on the parameters, or because the mean and higher moments do not exist when k ≤ − 1. The currently favoured estimators are those obtained by the method of probability‐weighted moments (PWM). The PWM estimators are good for cases where −1/2 < k < 1/2. Outside this range of k, the PWM estimates may not exist and if they do exist they cannot be recommended because their performance worsens as k increases. In this paper, we propose a method for estimating the parameters and quantiles of the GEVD. The estimators are well‐defined for all possible combinations of parameter and sample values. They are also easy to compute as they are based on equations which involve only one variable (rather than three). A simulation study is implemented to evaluate the performance of the proposed method and to compare it with the PWM. The simulation results seem to indicate that the proposed method is comparable to the PWM for −1/2 < k < 1/2 but outside this range it gives a better performance. Two real‐life environmental data sets are used to illustrate the methodology.