Moment Estimator-Based Extreme Quantile Estimation with Erroneous Observations: Application to Elliptical Extreme Quantile Region Estimation
Jaakko Pere, Pauliina Ilmonen, Lauri Viitasaari
TL;DR
The article tackles the impact of measurement or observation errors on moment-based extreme value estimators, focusing on the extreme value index $γ$ and corresponding quantiles across varied tail regimes. It develops error bounds under extended regular variation frameworks (ERV and 2ERV) and shows that, provided the observation error decays sufficiently fast relative to tail scales, standard asymptotic results still hold; it also addresses the delicate case $γ \le 0$ where errors can dominate. Extending these ideas to multivariate elliptical models, the paper constructs an affine-equivariant estimator for extreme quantile regions by replacing true location, scatter, and generating variates with estimators, and proves consistency in probability for the region in the symmetric difference sense. This work thus offers a rigorous treatment of approximation errors in high-dimensional extreme-value analysis and provides practical tools for affine-equivariant region estimation under elliptical distributions.
Abstract
In many application areas of extreme value theory, the variables of interest are not directly observable but instead contain errors. In this article, we quantify the effect of these errors in moment-based extreme value index estimation, and in corresponding extreme quantile estimation. We consider all, short-, light-, and heavy-tailed distributions. In particular, we derive conditions under which the error is asymptotically negligible. As an application, we consider affine equivariant extreme quantile region estimation under multivariate elliptical distributions.