Show that the regression function also has the following pointwise optimality property: E |m(X) Y | 2 |X = x = min f E |f(X) Y | 2 |X = x for -almost all x Rd. Let (X, Y ) be an RdR-valued random...

Show that the regression function also has the following pointwise optimality property:

E |m(X) − Y | 2 |X = x = min f E |f(X) − Y | 2 |X = x

for µ-almost all x ∈ Rd.

Let (X, Y ) be an Rd×R-valued random variable with E|Y | <>∗ : Rd → R which minimizes the L1 risk, i.e., which satisfies

E{|f ∗(X) − Y |} = min f:Rd→R E{|f(X) − Y |}.