[Paper Review] Elemental unbiased estimators for the Generalized Pareto tail

FA McRobie|arXiv (Cornell University)|Apr 14, 2013

Financial Risk and Volatility Modeling8 references3 citations

TL;DR

This paper introduces unbiased, location- and scale-invariant 'elemental' estimators for the Generalized Pareto Distribution (GPD) tail parameter using three log-spacings, achieving exact unbiasedness even at sample size N=3. It demonstrates that these elemental estimators form a complete basis for all such unbiased estimators constructed from linear combinations of log-spacings, with preliminary evidence suggesting they can be combined to form consistent estimators for pure GPD samples.

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.

Motivation & Objective

• To develop unbiased estimators for the GPD tail parameter that are invariant under location and scale transformations.
• To construct estimators based solely on log-spacings that maintain unbiasedness even for very small sample sizes, such as N=3.
• To establish whether elemental estimators—built from three log-spacings—form a complete basis for all possible unbiased, location- and scale-invariant estimators derived from linear combinations of log-spacings.
• To explore the potential of combining elemental estimators to achieve consistency in estimating the GPD tail parameter for samples from the pure GPD family.

Proposed method

• The method constructs estimators as linear combinations of three log-spacings derived from order statistics of the sample.
• It ensures unbiasedness by enforcing constraints that eliminate location and scale dependencies while maintaining exact unbiasedness for finite samples.
• The approach leverages the invariance properties of log-spacings under location and scale shifts to derive estimators that are invariant by construction.
• A basis of elemental estimators is derived, representing the minimal set of building blocks for all unbiased, location- and scale-invariant estimators in the log-spacing framework.
• Theoretical derivation confirms that these elemental estimators span the entire space of such unbiased estimators.
• Preliminary numerical experiments are conducted to assess whether linear combinations of elementals yield consistent estimators for pure GPD data.

Experimental results

Research questions

• RQ1 Can unbiased, location- and scale-invariant estimators for the GPD tail parameter be constructed that are exactly unbiased even at sample size N=3?
• RQ2 Do the elemental estimators—each based on three log-spacings—form a complete basis for all unbiased, location- and scale-invariant estimators built from linear combinations of log-spacings?
• RQ3 Is it possible to combine elemental estimators to produce a consistent estimator for the GPD tail parameter when samples are drawn from the pure GPD distribution?
• RQ4 What is the theoretical and empirical behavior of elemental estimators in finite-sample settings, particularly for small N?

Key findings

• The elemental estimators are exactly unbiased for the GPD tail parameter at sample sizes as small as N=3, demonstrating finite-sample unbiasedness.
• The set of elemental estimators forms a complete basis for all unbiased, location- and scale-invariant estimators constructed from linear combinations of log-spacings.
• Theoretical analysis confirms that no other such estimators exist outside the span of these elemental forms.
• Preliminary numerical results suggest that specific linear combinations of elemental estimators can yield consistent estimators for samples from the pure GPD family.
• The construction relies entirely on the structure of log-spacings, which inherently encode scale- and location-invariant information.
• The method provides a systematic framework for generating unbiased estimators without requiring asymptotic approximations.