Prove that for d ≤ 2 Lemma 6.4 is not distribution-free, i.e., construct a distribution of X for which Lemma 6.4 does not hold. Hint: Put d = 1 and assume a density f(x)=3x2, then F(x) = x3 and...

Prove that for d ≤ 2 Lemma 6.4 is not distribution-free, i.e., construct a distribution of X for which Lemma 6.4 does not hold. Hint: Put d = 1 and assume a density f(x)=3x2, then F(x) = x3 and

E{X(1,n)(X) − X2 } ≥

                                    ≥ C n5/3 .