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:

$\begin{array}{rl}& (1-\tau )\underset{-\mathrm{\infty }}{\overset{t}{\int }}(t-x)dF(x)=\tau \underset{t}{\overset{\mathrm{\infty }}{\int }}(x-t)dF(x)\\ & \underset{-\mathrm{\infty }}{\overset{t}{\int }}|t-x|dF(x)=\tau \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|x-t|dF(x)\\ & t-\mathrm{E}[X]=\frac{2\tau -1}{1-\tau }\underset{t}{\overset{\mathrm{\infty }}{\int }}(x-t)dF(x)\end{array}$

Quantile regression minimizes an asymmetric $L_1$ loss (see least absolute deviations). Analogously, expectile regression minimizes an asymmetric $L_2$ loss (see ordinary least squares):

$\begin{array}{rl}\mathrm{quantile}(\tau )& \in {\mathrm{argmin}}_{t\in \mathbb{ℝ}}\mathrm{E}[|X-t||\tau -H(t-X)|]\\ \mathrm{expectile}(\tau )& \in {\mathrm{argmin}}_{t\in \mathbb{ℝ}}\mathrm{E}[|X-t{|}^2|\tau -H(t-X)|]\end{array}$

where $H$ is the Heaviside step function.