Elemental estimators for the Generalized Extreme Value tail

TL;DR

This paper extends the concept of elemental estimators from the Generalized Pareto distribution to the Generalized Extreme Value (GEV) distribution, providing nearly unbiased estimators with small bias for tail analysis.

Contribution

It introduces elemental estimators for the GEV distribution, demonstrating their small bias and potential for constructing efficient estimators.

Findings

Elemental estimators have very small bias for GEV tails.

They are valid for all sample sizes, even as small as 3.

These estimators can serve as a basis for efficient tail estimators.

Abstract

In a companion paper (McRobie(2013) arxiv:1304.3918), a simple set of `elemental' estimators was presented for the Generalized Pareto tail parameter. Each elemental estimator: involves only three log-spacings; is absolutely unbiased for all values of the tail parameter; is location- and scale-invariant; and is valid for all sample sizes $N$, even as small as $N= 3$. It was suggested that linear combinations of such elementals could then be used to construct efficient unbiased estimators. In this paper, the analogous mathematical approach is taken to the Generalised Extreme Value (GEV) distribution. The resulting elemental estimators, although not absolutely unbiased, are found to have very small bias, and may thus provide a useful basis for the construction of efficient estimators.