Prove that for fixed k the k-NN regression estimate is weakly consistent for noiseless observations. Hint: See Problem 4.5. Show that E 1 k k i=1 (Y(i,n)(x) − m(X(i,n)(x)))2 µ(dx) = 1 k2 k i=1 E{σ2...

Prove that for fixed k the k-NN regression estimate is weakly consistent for noiseless observations. Hint: See Problem 4.5.

Show that

1 k k i=1 (Y(i,n)(x) − m(X(i,n)(x)))2 µ(dx) = 1 k2 k i=1 E{σ2 (X(i,n)(X))} → E(Y − m(X))2 k .