Computing Maximum Likelihood Estimates for the Generalized Pareto Distribution

The generalized Pareto distribution (GPD) is a two-parameter family of distributions that can be used to model exceedances over a threshold. Maximum likelihood estimators of the parameters are preferred, since they are asymptotically normal and asymptotically efficient in many cases. Numerical methods are required for maximizing the log-likelihood, however. This article investigates the properties of a reduction of the two-dimensional numerical search for the zeros of the log-likelihood gradient vector to a one-dimensional numerical search. An algorithm for computing the GPD maximum likelihood estimates based on this dimension reduction and properties are given.