TY - JOUR
T1 - One Axiom to Rule Them All: A Minimalist Axiomatization of Quantiles
AU - Fadina, Tolulope
AU - Liu, Peng
AU - Wang, Ruodu
N1 - Funding Information:
*Received by the editors October 31, 2022; accepted for publication (in revised form) February 21, 2023; published electronically June 1, 2023. https://doi.org/10.1137/22M1531567 Funding: The third author acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823, RGPAS-2018-522590). \dagger Department of Mathematical Sciences, University of Essex, Colchester, Essex CO4 3SQ, UK (t.fadina@ essex.ac.uk, [email protected]). \ddagger Department of Statistics and Actuarial Science, University of Waterloo, Waterloo N2L 3G1, ON, Canada ([email protected]).
Publisher Copyright:
© 2023 Society for Industrial and Applied Mathematics Publications. All rights reserved.
PY - 2023
Y1 - 2023
N2 - 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, elicitability, and locality 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.
AB - 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, elicitability, and locality 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.
KW - median
KW - ordinality
KW - quantile maximization
KW - quantiles
KW - Value-At-Risk
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U2 - 10.1137/22M1531567
DO - 10.1137/22M1531567
M3 - Article
AN - SCOPUS:85164420078
SN - 1945-497X
VL - 14
SP - 644
EP - 662
JO - SIAM Journal on Financial Mathematics
JF - SIAM Journal on Financial Mathematics
IS - 2
ER -