Check loss (${\rho }_{\tau }(u)=u(\tau -1_{\{u<0\}})$)

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.

For an integrable response with continuous strictly increasing distribution function $F$, the population risk for the check loss has derivative $F(q)-\tau$. Bounded difference quotients justify differentiation by dominated convergence. Integrating the derivative between the unique quantile and another location proves strict optimality. This proves identifiability without requiring a positive density.