On some methods of fitting the generalized Pareto distribution

Abstract

The two-parameter generalized Pareto distribution (GPD) has been recommended for the frequency analysis of environmental extreme events. In the present paper, we concentrate on one form of the GPD (which we will call GPD B) which can be useful in the frequency analysis of two types of hydrological variables: (1) when the shape parameter α is positive, the distribution (denoted by GPDB-2) can be used to study phenomena such as flood flows, which are bounded from below but have a long right tail; (2) when α is negative, the resulting distribution (GPD B-3) could be used to study variables such as low flows which are bounded from both sides but with a left tail. Six versions of the generalized method of moments (GMM) for fitting GPD B are investigated. The flexibility of the GMM provides the user with the possibility of choosing a version of the method (i.e. the moment pair that is used in fitting the distribution) which assigns larger weight to the larger elements of the sample, or another version which gives more weight to the smaller elements, depending on the problem at hand. A general formula for the asymptotic variance of the T-year event X T obtained by combining any two moments of GPD B is presented and applied. It is shown that the adequate choice of the order of these two moments to fit the distribution can lead, in some cases, to a considerable reduction in the variance of the estimator of X T , in comparison with estimation by the traditional method of moments (which uses moments of order one and two). The performance indices that are used to compare the different versions of the GMM are based on root mean square error, bias, and variance—both asymptotic and observed (based on simulation)—of GPD B quantiles and parameters. It is shown that moments of order ( 0 ),−1) and (2,−1) lead to the best results when the shape parameter a is positive (GPD B − 2), and the traditional method of moments can be considered as most efficient for negative values of α (GPD B −3). The GMM with moments of order 0 and one is shown to be rather consistent and moderately satisfactory for both GPD B-2 and GPD B-3.