Bias and size corrections in extreme value modeling

ABSTRACT

Extreme value theory models have found applications in myriad fields. Maximum likelihood (ML) is attractive for fitting the models because it is statistically efficient and flexible. However, in small samples, ML is biased to O(N⁻¹) and some classical hypothesis tests suffer from size distortions. This paper derives the analytical Cox–Snell bias correction for the generalized extreme value (GEV) model, and for the model's extension to multiple order statistics (GEVr). Using simulations, the paper compares this correction to bootstrap-based bias corrections, for the generalized Pareto, GEV, and GEVr. It then compares eight approaches to inference with respect to primary parameters and extreme quantiles, some including corrections. The Cox–Snell correction is not markedly superior to bootstrap-based correction. The likelihood ratio test appears most accurately sized. The methods are applied to the distribution of geomagnetic storms.

KEYWORDS:

• Extreme value theory
• Generalized Pareto distribution
• Generalized extreme value distribution
• Cox–Snell
• Small-sample bias
• Geomagnetic disturbances

Notes

1 The discrepancy became apparent in replication and one of the authors confirmed it.