Prove (5.7). Hint: Step (a). Let A be the event that X1, X2 ∈ [h/2, 3h/2], X3,...,Xn ∈ [5h/2, 1], and Y1 = Y2. Then P{A} > 0 and                 E  (mn(x)−m(x))2 µ(dx) = E  mn(x) 2 dx ≥  E...

Prove (5.7).

Hint: Step (a). Let A be the event that X1, X2 ∈ [h/2, 3h/2], X3,...,Xn ∈ [5h/2, 1], and Y1 = Y2. Then P{A} > 0 and

                E  (mn(x)−m(x))2 µ(dx) = E  mn(x) 2 dx ≥  E

Step (b). Given A, on [h/2, 3h/2], the locally linear kernel estimate mn has the form

                    mn(x) = ±2 ∆ (x − c),

where ∆ = |X1 − X2| and h/2 ≤ c ≤ 3h/2. Then

                      E

Step (c).

                     E  1 ∆|A  = E  1 ∆ = ∞.