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Statistical Inference for the Generalized Pareto Distribution: Maximum Likelihood Revisited

The maximum likelihood system of equations for the shape and scale parameters ξ and β of a generalized Pareto distribution is summarized through a single equation h x (ρ) = 0 where x designates the n-sample of realizations and . The study of the function h x (ρ) in the neighborhood of 0 puts forward a simple statistic S(x) only based on the first two empirical moments which is independent from β. This statistic allows to build an immediate and easy to use statistical test for ξ. The power of this test appears as quite similar to the one of already known test procedures which are much more difficult to use in practice. Moreover, the maximum likelihood solution of h x (ρ) = 0, say ρˆ, and S(x) both have the same sign. This property allows us to propose a controlled use of standard optimization algorithms for characterizing ρˆ, and as a consequence, the maximum likelihood estimations of ξ and β, without any risk of convergence failure or aberrant solution.