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

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

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):

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

where $H$ is the Heaviside step function.