We give an axiomatic foundation to $\Lambda$-quantiles, a family of generalized quantiles introduced by Frittelli et al. (2014) under the name of Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by a property that we call "locality", that means that any change in the distribution of the probability mass that arises entirely above or below the value of the $\Lambda$-quantile does not modify its value. We compare with a related axiomatization of the usual quantiles given by Chambers (2009), based on the stronger property of "ordinal covariance", that means that quantiles are covariant with respect to increasing transformations. Further, we present a systematic treatment of the properties of $\Lambda$-quantiles, refining some of the results of Frittelli et al. (2014) and Burzoni et al. (2017) and showing that in the case of a nonincreasing $\Lambda$ the properties of $\Lambda$-quantiles closely resemble those of the usual quantiles.

翻译：我们给Frittelli等人(2014年)以Lambda值为名推出的通用量子系(Lambda$-quantiles)提供了一种直观基础。根据温和假设,我们证明这些功能的特性具有我们称之为“地方性”的属性,这意味着,完全高于或低于美元值的概率质量分布的任何变化都不会改变其价值。我们比较了Corporation(2009年)根据“正常共变”这一较强的属性给出的通常量子的相关异质化(2009年),这意味着量子在不断增长的变异方面具有共变异性。此外,我们系统地处理了“Lambda$-quantiles”的属性,完善了Frittelli等人(2014年)和Burzoni等人(2017年)的一些结果,并表明,在不增加美元的情况下,美元-Lambda$-qontiles的属性与通常的相似。