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: ^{[1] [2] [3]}

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

Quantile regression minimizes an asymmetric ${\displaystyle L_{1}}$ loss (see least absolute deviations). Analogously, expectile regression minimizes an asymmetric ${\displaystyle L_{2}}$ loss (see ordinary least squares):

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

where ${\displaystyle H}$ is the Heaviside step function.

References

1. Werner Ehm, Tilmann Gneiting, Alexander Jordan, Fabian Krüger, "Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations, and Forecast Rankings," arxiv
2. Yuwen Gu and Hui Zou, "Aggregated Expectile Regression by Exponential Weighting," Statistica Sinica, https://www3.stat.sinica.edu.tw/preprint/SS-2016-0285_Preprint.pdf
3. Whitney K. Newey, "Asymmetric Least Squares Estimation and Testing," Econometrica, volume 55, number 4, pp. 819–47.