Quantiles for multivariate distribution: a short survey of population concepts
Abstract
Quantiles are among the most fundamental concepts in Probability and Statistics, from descriptive to inferential. However, quantile functions are well-defined and well-understood in the context of one-dimensional probability distributions, where their definition as the inverse of distribution functions---a definition which is intimately related to the canonical ordering of the real line. Starting with dimension d=2, such a canonical ordering is no longer available in~ℝd; nor is it available in nonlinear manifolds such as hyperspheres, tori, or polyspheres (vectors of directional variables). This results in the absence of an obvious and widely accepted definition of quantiles. The need to extend the concept of quantile beyond the classical univariate context nevertheless has sparked a large body of literature, giving rise to a variety of more or less satisfactory quantile concepts; the recent years have been particularly active in this respect. This short review is an attempt to complement and update Serfling's 25-year-old survey (Serfling 2008) by summarizing and unifying some of these concepts in light of recent contributions.
Related
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- A Uniformly Efficient Rejection Sampler for the Multivariate Expectile-Based Distribution
- A Mathai-type Multivariate Nakagami-m Distribution: Density, Closed-form Estimators, and Asymptotic Efficiency
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BibTeX
@misc{hallin2026quantiles,
title = {{Quantiles for multivariate distribution: a short survey of population concepts}},
author = {Marc Hallin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07271},
eprint = {2609.07271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07271},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Quantiles for multivariate distribution: a short survey of population concepts
AU - Marc Hallin
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07271
UR - https://arxiv.org/abs/2609.07271
AB - Quantiles are among the most fundamental concepts in Probability and Statistics, from descriptive to inferential. However, quantile functions are well-defined and well-understood in the context of one-dimensional probability distributions, where their definition as the inverse of distribution functions---a definition which is intimately related to the canonical ordering of the real line. Starting with dimension d=2, such a canonical ordering is no longer available in~ℝ^d; nor is it available in nonlinear manifolds such as hyperspheres, tori, or polyspheres (vectors of directional variables). This results in the absence of an obvious and widely accepted definition of quantiles. The need to extend the concept of quantile beyond the classical univariate context nevertheless has sparked a large body of literature, giving rise to a variety of more or less satisfactory quantile concepts; the recent years have been particularly active in this respect. This short review is an attempt to complement and update Serfling's 25-year-old survey (Serfling 2008) by summarizing and unifying some of these concepts in light of recent contributions.
ER -