pinball loss

The typical loss used when attempting to estimate quantiles. For a target quantile $q$, the loss is

$$L_q(\tau ,y)=\{\begin{array}{ll}(y-\tau )q,& y>\tau ,\\ (\tau -y)(1-q),& y\le \tau .\end{array}$$

Here $\tau$ should be interpreted as your guess for the $q$-th quantile, and $y$ is the observed value. In the case where $q=1/2$ we have

$$L_{1/2}(\tau ,y)=\frac{1}{2}|\tau -y|,$$

which is just the ${\ell }_1$ loss, the minimizer of which the median.

Suppose $q=0.9$. Then if $y>\tau$ we are “surprised” in some sense, since this should not happen very often. So the loss takes on a higher value (proportional to $q$). But if $y\le \tau$ we are less surprised, so the loss is small. The minimizer of the loss can be shown to be exactly the $q$ quantile.