This exercise involves some of the details in the sketches of the proof of Theorem 16.14 outlined below the theorem.

(a) Show that cj = OP(n), j = 0, 1.

(b) Verify the expression var(Δ|Z) = 2σ2

0tr(DV1,0DV1,0).

(c) Show that the three terms on the right side of (16.62) are OP(n

−1).

(d) Show that var(Δ|Z) = OP(n

−1) implies that Δ2 = oP(1).