Elemental unbiased estimators for the Generalized Pareto tail

TL;DR

This paper introduces unbiased, location- and scale-invariant elemental estimators for the GPD tail parameter, which are valid for small sample sizes and form a complete basis for such estimators, with preliminary evidence of consistency.

Contribution

The paper develops a new class of unbiased elemental estimators for the GPD tail parameter, complete with a basis and initial consistency evidence.

Findings

Estimators are unbiased for small samples as small as N=3.

Elementals form a complete basis for linear combinations of log-spacings.

Preliminary numerical results suggest consistency of elemental combinations.

Abstract

Unbiased location- and scale-invariant `elemental' estimators for the GPD tail parameter are constructed. Each involves three log-spacings. The estimators are unbiased for finite sample sizes, even as small as N=3. It is shown that the elementals form a complete basis for unbiased location- and scale-invariant estimators constructed from linear combinations of log-spacings. Preliminary numerical evidence is presented which suggests that elemental combinations can be constructed which are consistent estimators of the tail parameter for samples drawn from the pure GPD family.