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 , the expectile at level of the probability distribution with cumulative distribution function is uniquely characterized by any of the following equivalent conditions: 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.
Quantile regression minimizes an asymmetric loss (see least absolute deviations):
where is the Heaviside step function; analogously, expectile regression minimizes an asymmetric loss (see ordinary least squares):
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