Suppose that the loss function is the piecewise linear function

\[ \operatorname{Loss}(y, \hat{y})=\alpha(\hat{y}-y)_{+}+\beta(y-\hat{y})_{+}, \quad \alpha, \beta>0 \]

where \(c_{+}\)is equal to \(c\) if \(c>0\), and zero otherwise. Show that the minimizer of the risk \(\ell(g)=\mathbb{E} \operatorname{Loss}(Y, g(\boldsymbol{X}))\) satisfies

\[ \mathbb{P}\left[Y

In other words, \(g^{*}(\boldsymbol{x})\) is the \(\beta /(\alpha+\beta)\) quantile of \(Y\), conditional on \(\boldsymbol{X}=\boldsymbol{x}\).