Fitting the generalized Pareto distribution to data

The generalized Pareto distribution (GPD) was introduced by Pickands to model exceedances over a threshold. It has since been used by many authors to model data in several fields. The GPD has a scale parameter (sigma > 0) and a shape parameter (-infinity < k < infinity). The estimation of these parameters is not generally an easy problem. When k > 1, the maximum likelihood estimates do not exist, and when k is between 1/2 and 1, they may have problems. Furthermore, for k less than or equal to -1/2, second and higher moments do not exist, and hence both the method-of-moments (MOM) and the probability-weighted moments (PWM) estimates do not exist. Another and perhaps more serious problem with the MOM and PWM methods is that they can produce nonsensical estimates (i.e., estimates inconsistent with the observed data). In this article we propose a method for estimating the parameters and quantiles of the GPD. The estimators are well defined for all parameter values. They are also easy to compute. Some asymptotic results are provided. A simulation study is carried out to evaluate the performance of the proposed methods and to compare them with other methods suggested in the literature. The simulation results indicate that although no method is uniformly best for all the parameter values, the proposed method performs well compared to existing methods. The methods are applied to real-life data. Specific recommendations are also given.