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 $τ \in (0, 1)$ , the expectile $t$ at level $τ$ of the probability distribution with cumulative distribution function $F$ is uniquely characterized by any of the following equivalent conditions:

\begin{matrix} (1 - τ) \int_{- \infty}^{t} (t - x) d F (x) = τ \int_{t}^{\infty} (x - t) d F (x); \\ \int_{- \infty}^{t} | t - x | d F (x) = τ \int_{- \infty}^{\infty} | x - t | d F (x); \\ t - E [X] = \frac{2 τ - 1}{1 - τ} \int_{t}^{\infty} (x - t) d F (x) . \end{matrix}

Quantile regression minimizes an asymmetric $L_{1}$ loss (see least absolute deviations):

\begin{matrix} quantile (τ) & \in {argmin}_{t \in ℝ} E [| X - t | | τ - H (t - X) |], \end{matrix}

where $H$ is the Heaviside step function; analogously, expectile regression minimizes an asymmetric $L_{2}$ loss (see ordinary least squares):

\begin{matrix} expectile (τ) & \in {argmin}_{t \in ℝ} E [| X - t |^{2} | τ - H (t - X) |] . \end{matrix}

References

↑ Werner Ehm, Tilmann Gneiting, Alexander Jordan, Fabian Krüger, "Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations, and Forecast Rankings," arxiv
↑ 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
↑ Whitney K. Newey, "Asymmetric Least Squares Estimation and Testing," Econometrica, volume 55, number 4, pp. 819–47.

[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.