On the Impact of Approximation Errors on Extreme Quantile Estimation with Applications to Functional Data Analysis

TL;DR

This paper investigates how approximation errors affect the estimation of extreme quantiles in heavy-tailed data, providing conditions under which classical estimators remain valid and introducing new concentration inequalities for order statistics.

Contribution

It offers novel conditions for approximation errors to preserve asymptotic properties of estimators and introduces the first Chernoff-type concentration inequality for order statistics.

Findings

Conditions established for approximation errors to maintain Hill estimator validity

Derived a new concentration inequality for order statistics with explicit convergence rates

Quantified the impact of discretization and regularity on functional data norms

Abstract

We study the effect of approximation errors in assessing the extreme behavior of heavy-tailed random objects. We give conditions for the approximation error such that the standard asymptotic results hold for the classical Hill estimator and the corresponding extreme quantile estimator. As an application, we consider the effect of discretization errors in the computation of the $L^p$-norms related to functional data. We approximate the norms both with Riemann sums and with Monte Carlo integration. We quantify connections between the number of observed functions, the number of discretization points, and the regularity of the underlying functions. In addition, we derive a new concentration inequality for order statistics. This, to the best of our knowledge, is the first Chernoff-type concentration inequality for order statistics presented in the literature that provides an explicit rate at which the ratio between order statistics and tail quantile function converges to one. In our application, the bound is used to provide concentration inequalities measuring the distance between the Hill estimator based on approximated norms and the one based on the true ones.