Peter Thomson

SRA

Mixed methods for fitting the GEV distribution

The generalised extreme-value (GEV) distribution is widely used for modelling and characterising extremes. It is a flexible 3-parameter distribution that combines three extreme-value distributions within a single framework: the Gumbel, Frechet and Weibull. Common methods used for estimating the GEV parameters are the method of maximum likelihood and the method of L-moments.

This paper generalises the mixed maximum likelihood and L-moments GEV estimation procedures proposed by Morrison and Smith (2002) and derives the asymptotic properties of the resulting estimates. Analytic expressions are given for the asymptotic covariance matrices in a number of important cases, including the estimators proposed by Morrison and Smith (2002). These expressions are verified by simulation and the efficiencies of the various estimators established.

The asymptotic results are compared to those obtained for small samples, and the properties of the various estimators, including constrained maximum likelihood estimators, are considered. The corresponding quantile estimators are also assessed for accuracy and bias. Using simplified constraints for the support of the log-likelihood, computational strategies and graphical tools are developed which lead to computationally efficient, numerically robust, estimation procedures. These methods are also applied to 24-hour annual maximum rainfall at Wellington, New Zealand, over the period 1940-1949 and within each phase of the Interdecadal Pacific Oscillation (IPO).