Estimation and Inference of Discontinuity in Density

w.p.a.1., where the equality follows from a second-order expansion with _ on the line joining and 0,and the inequality follows from 1 P nnh i=1 g i (^) g i (^) 0 p! V , boundedness of V , and Lemma 2. Also notethatsup ^P (^; ) sup ^P ( 0 ; ) = O p(nh) 1 ; (24)2 n(^)2 n( 0 )w.p.a.1., where the inequality follows from the de…nition of ^, and the equality follows from Lemma 3with = 0 and 1 P nnh i=1 g i ( 0 ) = O p (nh) 1=2 (by Lemma 1). Since ~ 2 n , Lemma 2 guarantees~ 2 n (^), w.p.a.1., which implies ^P^; ~ sup ^P 2n(^) (^; ). Thus, combining (23) and (24),n j^gj Cn 2 O p(nh) 1 ; (25)w.p.a.1. Since we chose to satisfy (nh) 1=2 n ! 0, we have j^gj = O p n . Now, pick any n ! 0and de…ne = n^g. From j^gj = O p n , we have = o p n and 2 n n (^). Thus, we applythe same argument to (23)-(25) after replacing ~ with . Then we obtain n j^gj 2 C 2 n j^gj 2 ^P ^; sup ^P (^; ) = O p(nh) 1 ;2 n(^)which implies n j^gj 2 = O p(nh) 1 : Since this results holds for any n ! 0, we obtain the conclusion.20