Prove that, for σ2(x) ≥ c1 > 0,               Emn − m2 ≥ c2 nhd n with some constant c2 (cf. Beirlant and Gy¨orfi (1998)). Hint: Take the lower bound of the first term in the right-hand side in the...

Prove that, for σ2(x) ≥ c1 > 0,

              Emn − m2 ≥ c2 nhd n

with some constant c2 (cf. Beirlant and Gy¨orfi (1998)).

Hint: Take the lower bound of the first term in the right-hand side in the decomposition (4.4).

Let the random variable B(n, p) be binomially distributed with parameters n and p. Then

                    E 1 B(n, p) I{B(n,p)>0}  ≥ 1 np (1 − (1 − p) n) 2 .

 Hint: Apply the Jensen inequality