Probability-Matching Predictors for Extreme Extremes

TL;DR

This paper introduces a location- and scale-invariant predictor based on the Generalized Pareto Distribution that achieves accurate probability matching for extreme predictions across various distributions, even with small sample sizes.

Contribution

It presents a novel predictor that provides exact probability matching for GPD in extreme tail limits and approximates well for intermediate tail parameters, using hypergeometric functions and sampling theory.

Findings

Achieves exact probability matching in heavy-tailed and bounded-tail extremes.

Performs well even with small sample sizes as low as N=3.

Uses hypergeometric functions for derivation, inspired by Bayesian methods.

Abstract

A location- and scale-invariant predictor is constructed which exhibits good probability matching for extreme predictions outside the span of data drawn from a variety of (stationary) general distributions. It is constructed via the three-parameter {\mu, \sigma, \xi} Generalized Pareto Distribution (GPD). The predictor is designed to provide matching probability exactly for the GPD in both the extreme heavy-tailed limit and the extreme bounded-tail limit, whilst giving a good approximation to probability matching at all intermediate values of the tail parameter \xi. The predictor is valid even for small sample sizes N, even as small as N = 3. The main purpose of this paper is to present the somewhat lengthy derivations which draw heavily on the theory of hypergeometric functions, particularly the Lauricella functions. Whilst the construction is inspired by the Bayesian approach to the prediction problem, it considers the case of vague prior information about both parameters and model, and all derivations are undertaken using sampling theory.