Optimally robust estimators in generalized Pareto models

Abstract

In this paper, we study the robustness properties of several procedures for the joint estimation of shape and scale in a generalized Pareto model. The estimators that we primarily focus upon, most bias robust estimator (MBRE) and optimal MSE-robust estimator (OMSE), are one-step estimators distinguished as optimally robust in the shrinking neighbourhood setting; that is, they minimize the maximal bias, respectively, on such a specific neighbourhood, the maximal mean squared error (MSE). For their initialization, we propose a particular location–dispersion estimator, MedkMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally robust estimators are compared to the maximum-likelihood, skipped maximum-likelihood, Cramér–von-Mises minimum distance, method-of-medians, and Pickands estimators. To quantify their deviation from robust optimality, for each of these suboptimal estimators, we determine the finite-sample breakdown point and the influence function, as well as the statistical accuracy measured by asymptotic bias, variance, and MSE – all evaluated uniformly on shrinking neighbourhoods. These asymptotic findings are complemented by an extensive simulation study to assess the finite-sample behaviour of the considered procedures. The applicability of the procedures and their stability against outliers are illustrated for the Danish fire insurance data set from the package evir.

Keywords:

• generalized Pareto distribution
• robustness
• shrinking neighbourhood

AMS Subject Classifications:

• 62F10
• 62F35

Acknowledgements

We thank two anonymous referees for their valuable and helpful comments. This work was supported by a DAAD scholarship for N. Horbenko. It is part of her PhD thesis, a preprint of it is 42.

Notes

Due to the affine equivariance of the MBRE, OBRE, and OMSE in the location and scale setting, interpolation in package RobLox is done only for the varying radius r.