Lambda-quantiles under the microscope

2026-08-07

AI Summary

This study investigates Lambda quantiles under non-monotonic parameter functions Λ, addressing fundamental challenges in risk measurement concerning closure properties, identifiability, and structural complexity. By relaxing the conventional monotonicity assumption, the authors introduce weak lower semicontinuity and mixed convexity to establish a mixed representation theorem for Λ of bounded variation. They further propose an ordinal covariance group to characterize the symmetry and invariance inherent in such quantiles. Integrating tools from functional analysis, measure theory, and risk measure theory, the work reconstructs the correspondence between Λ and families of probability distributions, achieving near-complete identifiability of Λ. Notably, the bounded-variation case is reduced to the monotone setting, substantially streamlining the theoretical framework.

Abstract

We study Lambda-quantiles, a generalisation of classical quantiles in which the constant probability level $λ\in [0,1]$ is replaced by a functional parameter $Λ\colon \mathbb{R} \to [0,1]$. We consider the general case of non-monotone $Λ$, which arises naturally if closure properties of the class of corresponding Lambda-quantiles with respect to inf-aggregation or with respect to mixtures are required. As preliminary results, we characterise finiteness, constancy, and what we call the attainment property known from classical quantiles. We then consider the problem of reconstructing $Λ$ from the values of $Λ$-quantiles on a suitable family of simple distributions, showing its identifiability under mild assumptions. Next, we substantially refine several results obtained in the literature on weak upper and lower semicontinuity and on the property of convexity of the level sets with respect to mixtures, obtaining in both cases almost complete characterisations without any monotonicity assumption. We then move to the case in which $Λ$ has bounded variation, which enables us to prove a mixture representation result: any such $Λ$-quantile can be rewritten as a Lambda-quantile with an increasing functional parameter, evaluated at a mixture of the original distribution with a fixed reference distribution at a fixed weight, thus reducing the complexity of the parameter from bounded variation to monotone. Finally, we introduce and study the notion of the ordinal covariance group of a risk measure, showing that in the case of a $Λ$-quantile it coincides with the compositional invariance group of $Λ$ and with a certain group of measure-preserving transformations of the signed measure associated with $Λ$.

Problem

• Lambda-quantiles
• non-monotone
• identifiability
• bounded variation
• risk measures

Innovation

• Lambda-quantiles
• non-monotone Lambda
• mixture representation
• bounded variation
• ordinal covariance group