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.
is characterized by any of the following equivalent conditions:[1] [2] [3] Quantile regression minimizes an asymmetric
loss (see least absolute deviations).
Analogously, expectile regression minimizes an asymmetric
loss (see ordinary least squares): where