Let mn(x) be the k-NN regression estimate. Prove that, for fixed k,                               n→∞ E  (mn(x) − m(x))2 µ(dx) = E(Y − m(X))2 k for all distributions of (X, Y ) with EY 2 ...

Let mn(x) be the k-NN regression estimate. Prove that, for fixed k,

                              n→∞ E  (mn(x) − m(x))2 µ(dx) = E(Y − m(X))2 k

for all distributions of (X, Y ) with EY 2 <>

                mn(x) = 1 k k i=1 m(X(i,n)(x)) + 1 k k i=1 (Y(i,n)(x) − m(X(i,n)(x))).