Quantiles and expectiles, which are two important concepts and tools in tail risk measurements, can be regarded as an extension of median and mean, respectively. Both of these tail risk measurers can actually be embedded in a common framework of $L_p$ optimization with the absolute loss function ($p=1$) and quadratic loss function ($p=2$), respectively. When 0-1 loss function is frequently used in statistics, machine learning and decision theory, this paper introduces an 0-1 loss function based $L_0$ optimisation problem for tail risk measure and names its solution as modile, which can be regarded as an extension of mode. Mode, as another measure of central tendency, is more robust than expectiles with outliers and easy to compute than quantiles. However, mode based extension for tail risk measure is new. This paper shows that the proposed modiles are not only more conservative than quantiles and expectiles for skewed and heavy-tailed distributions, but also providing or including the unique interpretation of these measures. Further, the modiles can be regarded as a type of generalized quantiles and doubly truncated tail measure whcih have recently attracted a lot of attention in the literature. The asymptotic properties of the corresponding sample-based estimators of modiles are provided, which, together with numerical analysis results, show that the proposed modiles are promising for tail measurement.

翻译：分位数和期望分位数是尾部风险度量中两个重要的概念和工具，可分别视为中位数和均值的扩展。这两种尾部风险度量实际上可以分别嵌入到基于绝对值损失函数（p=1）和二次损失函数（p=2）的Lp优化框架中。鉴于0-1损失函数在统计学、机器学习及决策理论中广泛应用，本文引入一种基于0-1损失函数的L0优化问题用于尾部风险度量，并将其解命名为模数，可视为众数的扩展。众数作为另一种集中趋势度量，比期望分位数更具异常值鲁棒性，且计算优于分位数。然而，基于众数的尾部风险度量扩展尚属创新。本文证明，对于偏态和厚尾分布，所提出的模数不仅比分位数和期望分位数更为保守，而且能提供或包含这些度量的唯一解释。此外，模数可视为一类近年来文献中备受关注的广义分位数和双截断尾部度量。本文给出了相应基于样本的模数估计量的渐近性质，结合数值分析结果表明，所提出的模数在尾部度量中具有良好应用前景。