Check loss
= Check loss
{title2=$\rho_\tau(u)=u(\tau-\mathbf1_{\{u<0\}})$}
= Pinball loss
{synonym}
= Quantile loss
{synonym}
The <check loss> weights positive errors by $\tau$ and negative errors by $1-\tau$. It is convex, continuous and Lipschitz with constant $\max(\tau,1-\tau)$. At $\tau=1/2$ it is half the absolute error. Its expected value elicits a <quantile>: asymmetry determines which probability level the fitted location targets.