Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n⁻¹), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.

KEYWORDS:

• Maximum likelihood
• Bias reduction
• Extreme values
• Generalized Pareto distribution
• Peaks over threshold
• Parametric bootstrap.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62F10
• 62F40
• 62N02
• 62N05

Acknowledgment

We are extremely grateful to Scott Grimshaw for supplying the R code that we used to implement his maximum likelihood algorithm, and to Lief Bluck and Kees van Kooten for providing access to the computing resources needed to complete this study in a timely manner. We are also appreciative of the helpful comments, suggestions, and questions received from the referees, Hugh Chipman, Paul Della-Marta, Ruud Koning, Jacob Schwartz, and participants at the 2009 Hawaii International Conference on Statistics, Mathematics and Related Fields; the 2010 Joint Statistical Meetings; and seminars in the Departments of Economics at the Universities of Auckland and Waikato, and the Department of Mathematics and Statistics at the University of Victoria.

Funding

The second author acknowledges financial support from King's University College at the University of Western Ontario.