Improved inference for the generalized Pareto distribution

The generalized Pareto distribution is commonly used to model exceedances over a threshold. In this paper, we obtain adjustments to the generalized Pareto profile likelihood function using the likelihood function modifications proposed by Barndorff-Nielsen (Biometrika 70 (1983) 343365), Cox and Reid (J. R. Stat. Soc. Ser. B. Stat. Methodol. 55 (1993) 467471), Fraser and Reid (Utilitas Mathematica 47 (1995) 33-53), Fraser, Reid and Wu (Biometrika 86 (1999) 249-264) and Severini (Biometrika 86 (1999) 235-247). We consider inference on the generalized Pareto distribution shape parameter, the scale parameter being a nuisance parameter. Bootstrap-based testing inference is also considered. Monte Carlo simulation results on the finite sample performances of the usual profile maximum likelihood estimator and profile likelihood ratio test and also their modified versions is presented and discussed. The numerical evidence favors the modified profile maximum likelihood estimators and tests we propose. Finally, we consider two real datasets as illustrations.