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

( 1 − τ ) ∫ − ∞ t ( t − x ) d F ( x ) = τ ∫ t ∞ ( x − t ) d F ( x ) ∫ − ∞ t | t − x | d F ( x ) = τ ∫ − ∞ ∞ | x − t | d F ( x ) t − E ⁡ [ X ] = 2 τ − 1 1 − τ ∫ t ∞ ( x − t ) d F ( x ) {\displaystyle {\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 {\displaystyle L_{1}} loss (see least absolute deviations). Analogously, expectile regression minimizes an asymmetric L 2 {\displaystyle L_{2}} loss (see ordinary least squares):

quantile ⁡ ( τ ) ∈ argmin t ∈ R ⁡ E ⁡ [ | X − t | | τ − H ( t − X ) | ] expectile ⁡ ( τ ) ∈ argmin t ∈ R ⁡ E ⁡ [ | X − t | 2 | τ − H ( t − X ) | ] {\displaystyle {\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 {\displaystyle H} is the Heaviside step function.