Symmetry measures under additive distortion measurement errors

Abstract

Lemma 1 Suppose E(V |W = w) = m(w) and its derivatives up to second order are bounded for all w ∈ [a 1 , a 2 ]. E|V | 3 exists and sup w |v| s f (w, v)dv < ∞, where f (w, v) is the joint density of (W, V ). Suppose (W i , V i ), i = 1, 2, . . . n are independent and identically distributed (i.i.d.) samples from (W, V ). If condition (C3) holds true for kernel function K(t), and n 2ϵ−1 h → ∞ for ϵ < 1 − s −1 , we have sup w∈ [a1,a2]