This exercise is concerned with some of the arguments used in the proof of Theorem 12.1.

(i) Show that (12.14) holds under (12.10)–(12.12), where θn is defined by (12.13) and rn is uniformly ignorable compared with the first term as ˜ θ varies on ¯En = {˜ θ : |Pn( ˜ θ − θ)| = 1}.

(ii) Derive (12.15) using the Taylor expansion and the result of part (i) of the theorem.