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

Modile as a conservative tail risk measurer: the solution of an optimisation problem with 0-1 loss function