One Axiom To Rule Them All: A Minimalist Axiomatization of Quantiles

21 Pages Posted: 18 Nov 2021 Last revised: 10 Feb 2022

See all articles by Tolulope Fadina

Tolulope Fadina

University of Essex

Peng Liu

University of Essex

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: October 17, 2021

Abstract

We offer a minimalist axiomatization of quantiles among all real-valued mappings on a general set of distributions through only one axiom. This axiom is called ordinality: quantiles are the only mappings that commute with all increasing and continuous transforms. Other convenient properties of quantiles, monotonicity, semicontinuity, comonotonic additivity and elicitability in particular, follow from this axiom. Furthermore, on the set of convexly supported distributions, the median is the only mapping that commutates with all monotone and continuous transforms. On a general set of distributions, the median interval is pinned down as the unique minimal interval-valued mapping that commutes with all monotone and continuous transforms. Finally, our main result, put in a decision-theoretic setting, leads to a minimalist axiomatization of quantile preferences. In banking and insurance, quantiles are known as the standard regulatory risk measure Value-at-Risk (VaR), and thus, an axiomatization of VaR is obtained with only one axiom among law-based risk measures.

Keywords: quantiles; median; ordinality; quantile maximization; Value-at-Risk

JEL Classification: D81

Suggested Citation

Fadina, Tolulope and Liu, Peng and Wang, Ruodu, One Axiom To Rule Them All: A Minimalist Axiomatization of Quantiles (October 17, 2021). Available at SSRN: https://ssrn.com/abstract=3944312 or http://dx.doi.org/10.2139/ssrn.3944312

Tolulope Fadina

University of Essex ( email )

Wivenhoe Park
Colchester Essex CO4 3SQ
United Kingdom

Peng Liu

University of Essex ( email )

Wivenhoe Park
Colchester, CO4
United Kingdom

Ruodu Wang (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

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