Assume that the weights {Wn,i} are nonnegative and that the corresponding local averaging estimate is weakly universally consistent. Prove that (iii) in Theorem 4.1 is satisfied (Stone (1977)). Hint:...

Assume that the weights {Wn,i} are nonnegative and that the corresponding local averaging estimate is weakly universally consistent. Prove that (iii) in Theorem 4.1 is satisfied (Stone (1977)).

Hint: For any fixed x0 and a > 0 let f be a nonnegative continuous function which is 0 on Sx0,a/3 and is 1 on Sc x0,2a/3. Choose Y = f(X) = m(X), then

                     I{X∈Sx0,a/3} n i=1 Wn,i(X)f(Xi) ≥ I{X∈Sx0,a/3} n i=1 Wn,i(X)I{Xi−X>a} → 0

in probability, therefore, for any compact set B,

                       I{X∈B} n i=1 Wni(X)I{Xi−X>a} → 0

in probability.