Quantification of estimation instability and its application to threshold selection in extremes

Abstract

The main goal of this paper is to propose a measure which quantifies the instability of estimates over a range of chosen values of some other parameter. The measure is used to identify a region where the estimates are considered "stable". Methods for identifying stable regions are then developed. These methods are applied to threshold selection in an extreme value analysis context, where the perturbed Pareto distribution is fitted to observed relative excesses. As a result a more accurate estimator of the extreme value index is obtained. As a further application the instability measure is employed in second order parameter estimation, which leads to an adjustment of an existing estimator, having desirable properties as far as its role in threshold selection is concerned, as well as improved estimation of the second order parameter for large samples.