To show

Given: $$Loss(y, \hat{y}) = \alpha (\hat{y}-y)_+ + \beta (y-\hat{y})_+$$ Note: $c_+$ is equal to c if c > 0 and zero otherwise!

Show that the minimizer of the risk $l(g) = E Loss(Y, g(X))$ satisfies: $$P[Y<g^* (x)= | X = x] = \frac{\beta}{\alpha + \beta}$$ where $g^* (x)$ is the optimal prediction function, that is $g^* (x)=argmin_{g \in G} l(g)$ in function class $G$


I interpret that $g^* (x)$ is the $\frac{\beta}{\alpha + \beta}$ quantile of Y conditional on $X = x$.

I'm uncertain about how to set up the probability distribution correctly and where to go from there.

Please comment if something is unclear. I'm just trying to learn!


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