Prove Theorem 10.1. Hint: Proceed as in the proof of Theorem 10.3, but apply Problem 10.1 instead of Theorem 10.2
(Devroye, personal communication, 1998). Let n ∈ N be fixed. Let Fn be the set of all functions which are piecewise linear on a partition of [0, 1] consisting of intervals. Assume that [0, h1] is one of these intervals (h1 > 0). Show that the piecewise linear estimate
mn(·) = arg min f∈Fn 1 n n i=1 |f(Xi) − Yi| 2
satisfies
E |mn(x) − m(x)| 2 µ(dx) = ∞
if X is uniformly distributed on [0, 1], Y is {−1, 1}-valued with EY = 0, and X, Y are independent.
Step (a). Let A be the event that X1, X2 ∈ [0, h1], X3,...,Xn ∈ [h1, 1], and Y1 = Y2. Then P{A} > 0 and
E (mn(x)−m(x))2 µ(dx) = E mn(x) 2 dx ≥ E
|mn(x)|dx A P{A} 2 .
Step (b). Given A, on [0, h1] the piecewise linear estimate mn has the form mn(x) = ±2 ∆ (x − c), where ∆ = |X1 − X2| and 0 ≤ c ≤ h1. Then
E
|mn(x)|dx A ≥ E 2 ∆
h1 0 |x − h1/2|dx A .
Step (c).
E 1 ∆ A = E 1 ∆ = ∞.