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 expectile of the probability distribution with cumulative distribution function 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd61b1131669505fadc5c14ada4faf779ec82fe)
Quantile regression minimizes an asymmetric loss (see least absolute deviations). Analogously, expectile regression minimizes an asymmetric 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44fcc7c46d5848c72a6a27fbe121e8ea92fce29b)
where is the Heaviside step function.
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.