  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.
