A smooth conditional quantile estimator and related applications of conditional empirical processes

AbstractLet {(Xi, Yi); i = 1,2,…} be a sequence of i.i.d. r.v.'s and denote by m(y | x0), −∞ < y < ∞, the conditional distribution function of Y given X = x0, −∞ < x0 < ∞. In this paper we propose and discuss certain smooth variants (based both on single as well as double kernel weights) of the standard conditional quantile estimator mn−1(λ | x0), 0 < λ < 1, of m−1(λ | x0), where mn(y | x0) is a (kernel) estimator of m(y | x0). The weak convergence of the corresponding conditional quantile process is also established. The same methods are used to study a new estimator of the conditional density and a “robust” estimator of the regression function