Abstract

As analogs of quantiles, extremiles are coherent spectral risk measures

with explicit formulations and intuitive interpretations. Their inherent sensitivity to the magnitude of extreme outcomes makes them particularly suitable for

heavy-tailed data.

However, existing extremile estimation methods rarely exploit rich auxiliary covariate information, which limits their ability to capture

conditional extreme patterns and to extrapolate reliably at very high risk levels.

This paper proposes a new nonparametric framework for estimating conditional

extremiles in the presence of multiple covariates. By combining reproducing kernel Hilbert spaces (RKHS) with a quantile regression process approximation, our

method flexibly models the conditional extremile structure while enabling reliable

extrapolation for heavy-tailed distributions. We establish the non-asymptotic error bound for the estimation error, rigorously justifying its theoretical validity.

Simulation studies show that our approach outperforms existing competitors in

both efficiency and extrapolation accuracy in heavy-tailed settings. An empirical

application to large commercial banks further illustrates its practical value for

extreme risk measurement.

Information

Preprint No.SS-2025-0294
Manuscript IDSS-2025-0294
Complete AuthorsFang Chen, Caixing Wang
Corresponding AuthorsCaixing Wang
Emailswangcaixing96@gmail.com

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Acknowledgments

The authors thank the editor, the associate editor, and two anonymous referees for their constructive suggestions, which significantly improved this paper.

Supplementary Materials

The supplementary materials contain some useful lemmas and the detailed proofs of the main

results in this paper.

Supplementary materials are available for download.