Lambda分位数的微观研究
Lambda-quantiles under the microscope
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中文总结 AI 辅助
该研究推广经典分位数为Lambda分位数，针对非单调Λ等情况完成多项理论刻画，提出混合表示结果并定义风险测度的序协方差群，完善了分位数相关的理论成果。
中文摘要 AI 辅助
我们研究Lambda分位数，它是经典分位数的推广，其中常数概率水平λ∈[0,1]被函数参数Λ:ℝ→[0,1]取代。我们考虑非单调Λ的一般情况，当要求对应Lambda分位数类关于下确界聚合或混合的闭包性质时，这种情况自然出现。作为初步结果，我们刻画了经典分位数中已知的有限性、恒定性和所谓的可达性。然后我们考虑从适当简单分布族的Lambda分位数值重构Λ的问题，证明在温和假设下其可识别性。接下来，我们大幅改进了文献中关于弱上半连续性、弱下半连续性以及水平集关于混合的凸性性质的若干结果，在两种情况下均在无任何单调性假设下获得几乎完整的刻画。随后我们转向Λ有界变差的情况，这使我们能证明混合表示结果：任何此类Lambda分位数可重写为具有递增函数参数的Lambda分位数，该分位数在原始分布与固定参考分布以固定权重混合的分布上求值，从而将参数的复杂度从有界变差降至单调。最后，我们引入并研究风险测度的序协方差群概念，表明在Lambda分位数的情况下，它与Λ的成分不变群以及与Λ相关联的符号测度的某类保测变换群重合。
英文摘要
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 $Λ$.
发表机构
- University of Milano-Bicocca（米兰比可卡大学）
- University of Amsterdam（阿姆斯特丹大学）
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