Expectile

In the mathematical theory of probability, the expectiles of a probability distribution are related to the expected value of the distribution in a way analogous to that in which the quantiles of the distribution are related to the median.

For $ \tau \in (0,1) $ expectile of the probability distribution with cumulative distribution function $F$  is characterized by any of the following equivalent conditions:



\begin{align} & (1-\tau)\int^t_{-\infty}(t-x) \, dF(x) = \tau\int^\infty_t(x-t) \, dF(x) \\[5pt] & \int^t_{-\infty}|t-x| \, dF(x) = \tau\int^\infty_{-\infty}|x-t| \, dF(x) \\[5pt] & t-\operatorname E[X]=\frac{2\tau-1}{1-\tau} \int^\infty_t(x-t) \, dF(x) \end{align} $$