Sometimes rate of convergence results for nonparametric regression estimation are only shown for bounded |Y |, so it is reasonable to consider minimax lower bounds for such classes. Let D∗(p,C) be the...

Sometimes rate of convergence results for nonparametric regression estimation are only shown for bounded |Y |, so it is reasonable to consider minimax lower bounds for such classes. Let D∗(p,C) be the class of distributions of (X, Y ) such that:

(I’) X is uniformly distributed on [0, 1]d;

(II’) Y ∈ {0, 1} a.s. and P{Y = 1|X = x} = 1 − P{Y = 0|X = x} = m(x) for all x ∈ [0, 1]d; and

(III’) m ∈ F(p,C) and m(x) ∈ [0, 1] for all x ∈ [0, 1]d.

Prove that for the class D∗(p,C) the sequence

                 an = n− 2p 2p+d

is a lower minimax rate of convergence.