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Quantile regression at quantile $\tau$ minimizes the following "pinball" loss function, $L_{\tau}$, and elicits conditional quantile $\tau$.

$$ l_{\tau}(y_i, \hat y_i) = \begin{cases} \tau\vert y_i - \hat y_i\vert, & y_i - \hat y_i \ge 0 \\ (1 - \tau)\vert y_i - \hat y_i\vert, & y_i - \hat y_i < 0 \end{cases}\\ L_{\tau}(y, \hat y) = \sum_{i=1}^n l_{\tau}(y_i, \hat y_i) $$

When $\tau = 0.5$, this loss function is absolute loss.

If we generalize squared loss the way that absolute loss generalizes to the pinball loss, denotes as $L^{^*}_{\tau}$ below, what is elicited?

$$ l^{^*}_{\tau}(y_i, \hat y_i) = \begin{cases} \tau\left( y_i - \hat y_i\right)^2, & y_i - \hat y_i \ge 0 \\ (1 - \tau)\left( y_i - \hat y_i\right)^2, & y_i - \hat y_i < 0 \end{cases}\\ L^{^*}_{\tau}(y, \hat y) = \sum_{i=1}^n l^{^*}_{\tau}(y_i, \hat y_i) $$

When $\tau = 0.5$, this is just the usual squared loss that elicits conditional means. When $\tau\ne 0.5$, I am not sure what such a loss function would elicit.

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    $\begingroup$ I’ve heard of something called an expectile whose definition hasn’t quite made sense to me. Perhaps this is an equivalent definition, one that is easier for me to interpret. $\endgroup$
    – Dave
    Commented Jan 4 at 17:27
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    $\begingroup$ You are correct, the (conditional) expectile is elicited. Are you asking for a definition? $\endgroup$
    – picky_porpoise
    Commented Jan 4 at 18:11
  • $\begingroup$ @picky_porpoise A way to tie this to the "tail expectation" would be appreciated. $\endgroup$
    – Dave
    Commented Jan 4 at 19:18

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