The sample fraction in peaks-over-threshold problems where the second-order expansion is valid with specific reference to the generalized Pareto distribution

TL;DR

This paper derives an expression for the threshold percentile in peaks-over-threshold problems where the generalized Pareto distribution's second-order approximation holds, aiding in optimal sample fraction estimation for tail index inference.

Contribution

It provides a new formula for the threshold percentile ensuring the validity of the second-order approximation in heavy-tailed distribution analysis.

Findings

Derived explicit threshold percentile expression

Improved accuracy in tail index estimation

Enhanced understanding of second-order approximation validity

Abstract

In samples from a heavy-tailed distribution a second-order approximation is often use to approximate the tail function. Based on the parameters of the approximation, an optimal sample fraction can be estimated which is then used to estimate the index. Given that the observations are above a threshold and has an approximate generalized Pareto distribution, an expression is derived for the percentile above which the second-order approximation is valid