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2024 | OriginalPaper | Chapter

Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours

Authors : Marc Hallin, Dimitri Konen

Published in: Applications of Optimal Transport to Economics and Related Topics

Publisher: Springer Nature Switzerland

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Abstract

Quantiles are a fundamental concept in probability and theoretical statistics and a daily tool in their applications. While the univariate concept of quantiles is quite clear and well understood, its multivariate extension is more problematic. After half a century of continued efforts and many proposals, two concepts, essentially, are emerging: the so-called (relabeled) geometric quantiles, extending the characterization of univariate quantiles as minimizers of an L\(_1\) loss function involving the check functions, and the more recent center-outward quantiles based on measure transportation ideas. These two concepts yield distinct families of quantile regions and quantile contours. Our objective here is to present a comparison of their main theoretical properties and a numerical investigation of their differences.

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As we shall see, the existence and (almost everywhere) uniqueness, in dimension \(d=1\), of a monotone non-decreasing mapping (that is, the derivative, hence the gradient, of a convex function) pushing \(\textrm{U}_1\) forward to \(\mathrm P\) is a particular case of a famous and more general measure transportation result by [13].

Formally, for any compact subset \(K\subset {\mathbb R}^d\), there exist constants \(0<c^-_K\le c_K^+<\infty \) such that \(c_K^-\le \textrm{dP}/\mathrm{d\lambda }(\textbf{x})\le c_K^+\) for all \(\textbf{x}\in K\). This assumption is maintained here for clarity of exposition, but it can be relaxed: see [3, 4].

Note, however, that the collections of center-outward regions or center-outward contours alone (that is, without the mappings \(\textbf{F}^{\scriptscriptstyle \pm }_\textrm{P}\) or \(\textbf{Q}^{\scriptscriptstyle \pm }_\textrm{P}\) between \({\mathbb R}^d\) and \(\mathbb {B}_d\)) do not characterize \(\mathrm P\).

In the case of a genuine sample, N is factorized into \(N=n_Rn_S + n_0\) with \(n_0< \min (n_R, n_S)\). We refer to [9] for details on the choice of \(n_R\) and \(n_S\). Here, however, we are dealing with simulations, and can choose N such that \(n_0=0\).

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Title: Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours
Authors: Marc Hallin
Dimitri Konen
Publisher: Springer Nature Switzerland
Book: Applications of Optimal Transport to Economics and Related Topics
Print ISBN: 978-3-031-67769-4

Electronic ISBN: 978-3-031-67770-0

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-67770-0_6

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