Functional Extreme-PLS

TL;DR

This paper introduces a novel dimension reduction method for functional data in infinite-dimensional spaces, combining ideas from PLS and SIR to effectively capture tail-related information in extreme value analysis.

Contribution

It extends the Extreme-PLS approach to infinite-dimensional covariates, proposing new estimators with proven asymptotic properties under mild regular variation conditions.

Findings

Demonstrates consistency and convergence rates of the estimators.

Shows effectiveness in finite-sample synthetic data.

Validates approach on real financial data for extreme risk estimation.

Abstract

We propose an extreme dimension reduction method extending the Extreme-PLS approach to the case where the covariate lies in a possibly infinite-dimensional Hilbert space. The ideas are partly borrowed from both Partial Least-Squares and Sliced Inverse Regression techniques. As such, the method relies on the projection of the covariate onto a subspace and maximizes the covariance between its projection and the response conditionally to an extreme event driven by a random threshold to capture the tail-information. The covariate and the heavy-tailed response are supposed to be linked through a non-linear inverse single-index model and our goal is to infer the index in this regression framework. We propose a new family of estimators and show its asymptotic consistency with convergence rates under the model. Assuming mild conditions on the noise, most of the assumptions are stated in terms of regular variation unlike the standard literature on SIR and single-index regression. Finally, our results are illustrated on a finite-sample study with synthetic functional data as well as on real data from the financial realm, highlighting the effectiveness of the dimension reduction for estimating extreme risk measures.