Quantiles, expectiles and extremiles can be seen as concepts defined via an optimization problem, where this optimization problem is driven by two important ingredients: the loss function as well as a distributional weight function. This leads to the formulation of a general class of functionals that contains next to the above concepts many interesting quantities, including also a subclass of distortion risks. The focus of the paper is on developing estimators for such functionals and to establish asymptotic consistency and asymptotic normality of these estimators. The advantage of the general framework is that it allows application to a very broad range of concepts, providing as such estimation tools and tools for statistical inference (for example for construction of confidence intervals) for all involved concepts. After developing the theory for the general functional we apply it to various settings, illustrating the broad applicability. In a real data example the developed tools are used in an analysis of natural disasters.

翻译：分位数、期望分位数和极值分位数可视为通过优化问题定义的概念，该优化问题由两个关键要素驱动：损失函数和分布权重函数。这引出了一类广义泛函的表述，除上述概念外，该类泛函还包含许多有价值的量，包括扭曲风险的一个子类。本文重点在于为该类泛函开发估计量，并建立这些估计量的渐近一致性和渐近正态性。这一通用框架的优势在于可应用于极为广泛的概念范畴，从而为所有涉及的概念提供估计工具和统计推断工具（例如用于构建置信区间）。在建立通用泛函理论后，我们将其应用于多种场景，以展示其广泛的适用性。在真实数据案例中，所开发工具被用于自然灾害分析。