Multivariate M-extremiles
Abstract
Extremiles represent an important class of distributional functionals defined through weighted squared loss functions, providing a valuable alternative to quantiles and expectiles. To date, their development has remained confined to the univariate setting. Here we introduce a robust multivariate extension of extremiles for random vectors by combining a directional weighting mechanism with Huber's multidimensional loss function. The proposed definition yields a family of directional extremiles indexed by a tuning parameter controlling the degree of robustness to outliers, with the quadratic-loss geometric extremile arising as a limiting case. We prove their main theoretical properties and introduce the notion of extremile sets. From an inferential perspective, we develop an efficient two-step procedure based on iteratively reweighted least squares, and derive asymptotic properties using the theory of Z-estimators. The finite-sample behavior of the proposed methodology is investigated through simulation experiments under Gaussian, contaminated, and heavy-tailed settings. An application to daily returns of three major cryptocurrencies illustrates its practical utility on financial data.
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BibTeX
@misc{merlo2026multivariate,
title = {{Multivariate M-extremiles}},
author = {Luca Merlo and Valeria Bignozzi and Lea Petrella and Nicola Salvati},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25492},
eprint = {2610.25492},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25492},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multivariate M-extremiles
AU - Luca Merlo
AU - Valeria Bignozzi
AU - Lea Petrella
AU - Nicola Salvati
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25492
UR - https://arxiv.org/abs/2610.25492
AB - Extremiles represent an important class of distributional functionals defined through weighted squared loss functions, providing a valuable alternative to quantiles and expectiles. To date, their development has remained confined to the univariate setting. Here we introduce a robust multivariate extension of extremiles for random vectors by combining a directional weighting mechanism with Huber's multidimensional loss function. The proposed definition yields a family of directional extremiles indexed by a tuning parameter controlling the degree of robustness to outliers, with the quadratic-loss geometric extremile arising as a limiting case. We prove their main theoretical properties and introduce the notion of extremile sets. From an inferential perspective, we develop an efficient two-step procedure based on iteratively reweighted least squares, and derive asymptotic properties using the theory of Z-estimators. The finite-sample behavior of the proposed methodology is investigated through simulation experiments under Gaussian, contaminated, and heavy-tailed settings. An application to daily returns of three major cryptocurrencies illustrates its practical utility on financial data.
ER -