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

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 applica...

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Bibliographic Details
Published in:arXiv.org 2024-10
Main Authors: Pere, Jaakko, Avelin, Benny, Garino, Valentin, Ilmonen, Pauliina, Viitasaari, Lauri
Format: Article
Language:English
Subjects:
Approximation
Continuity (mathematics)
Discretization
Errors
Extreme values
Norms
Quantiles
Random variables
Stochastic processes
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Summary: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.
ISSN:2331-8422