Variance-reduced extreme value index estimators using control variates in a semi-supervised setting

TL;DR

This paper introduces a semi-supervised, control variates-based transfer learning method to reduce variance in Extreme Value Index estimation, leveraging source data and tail dependence, with theoretical and practical validation.

Contribution

It proposes a novel variance reduction technique for EVI estimators using control variates in a semi-supervised setting, effective even with dissimilar tail heaviness.

Findings

Significant variance reduction proportional to tail dependence.

Method extends to other ratio-based EVI estimators.

Validated on water surge and ice accretion datasets.

Abstract

The estimation of the Extreme Value Index (EVI) is fundamental in extreme value analysis but suffers from high variance due to reliance on only a few extreme observations. We propose a control variates based transfer learning approach in a semi-supervised framework, where a small set of coupled target and source observations is combined with abundant unpaired source data. By expressing the Hill estimator of the target EVI as a ratio of means, we apply approximate control variates to both numerator and denominator, with jointly optimized coefficients that guarantee variance reduction without introducing bias. We show theoretically and through simulations that the asymptotic relative variance reduction of the transferred Hill estimator is proportional to the tail dependence between the target and source variables and independent of their EVI values. Thus, substantial variance reduction can be achieved even without similarity in tail heaviness of the target and source distributions. The proposed approach can be extended to other EVI estimators expressed with ratio of means, as demonstrated on the moment estimator. The practical value of the proposed method is illustrated on multi-fidelity water surge and ice accretion datasets.