On Uniform Confidence Intervals for the Tail Index and the Extreme Quantile

TL;DR

This paper investigates the challenges of constructing uniform confidence intervals for tail index and extreme quantiles, proving an impossibility result and proposing honest intervals that account for worst-case bias, validated through simulations and real data.

Contribution

It establishes an impossibility result for length-optimal uniform confidence intervals and introduces a method for honest intervals that incorporate worst-case bias.

Findings

Impossibility of length-optimal uniform confidence intervals under certain conditions

Proposed honest confidence intervals are valid across a non-parametric family

Method successfully applied to simulated and real-world data sets

Abstract

This paper presents two results concerning uniform confidence intervals for the tail index and the extreme quantile. First, we show that it is impossible to construct a length-optimal confidence interval satisfying the correct uniform coverage over a local non-parametric family of tail distributions. Second, in light of the impossibility result, we construct honest confidence intervals that are uniformly valid by incorporating the worst-case bias in the local non-parametric family. The proposed method is applied to simulated data and a real data set of National Vital Statistics from National Center for Health Statistics.