ON SOME METHODS OF FITTING THE GENERALIZED PARETO DISTRIBUTION

The two-parameter generalized Pareto distribution (GPD) has been recom mended for the frequency analysis of environmental extreme events, In the present paper, we concentrate on one form of the GPD (which Re wil l call GPD(B)) which can be useful in the Frequency analysis of two ty pes of hydrological variables: (1) when the shape parameter ct is posi tive, the distribution (denoted by GPD(B)-2) can be used to study phen omena such as flood flows, which are bounded from below but have a lon g right tail; (2) when alpha 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 gen eralized method of moments (GMM) for fitting GPD(B) are investigated. The flexibility of the GMM provides the user with the possibility of c hoosing a version of the method (i.e. the moment pair that is used in fitting the distribution) which assigns larger weight to the larger el ements of the sample. or ana ther version which gives more weight to t he smaller elements, depending on the problem at hand. A general formu la for the asymptotic variance of the T-year event X(T) Obtained by co mbining any two moments of GPD(B) is presented and applied, It is show n that the adequate choice of the order of these two moments to fit th e 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 differe nt 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 alpha is positive (GPD(B) - 2), and the traditional method of moments can be considered as most efficient for negative values of alpha (GPD(B) - 3 ). The GMM with moments of order 0 and one is shown to be rather consi stent and moderately satisfactory for both GPD(B)-2 and GPD(B)-3.