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 $\tau \in (0,1)$, the expectile of the probability distribution with cumulative distribution function $F$ is characterized by any of the following equivalent conditions: Whitney K. Newey, "Asymmetric Least Squares Estimation and Testing," ''Econometrica'', volume 55, number 4, pp. 819–47.

:
\begin
& (1-\tau)\int^t_(t-x) \, dF(x) = \tau\int^\infty_t(x-t) \, dF(x) \\ pt& \int^t_, t-x, \, dF(x) = \tau\int^\infty_, x-t, \, dF(x) \\ pt& t-\operatorname E \frac \int^\infty_t(x-t) \, dF(x)
\end

Quantile regression minimizes an asymmetric $L_1$ loss (see least absolute deviations).
Analogously, expectile regression minimizes an asymmetric $L_2$ loss (see ordinary least squares):
:
\begin
\operatorname(\tau) &\in \operatorname_ \operatorname X - t, ^2 , \tau - H(t - X), \end

where $H$ is the Heaviside step function.

References

{{reflist

Theory of probability distributions>X - t, , \tau - H(t - X), \\
\operatorname(\tau) &\in \operatorname_ \operatorname X - t, ^2 , \tau - H(t - X), \end

where $H$ is the Heaviside step function.

References

{{reflist

Theory of probability distributions