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

Let {(Xi, Yi); I = 1,2,...} be a sequence of i.i.d. r.v.'s and denote by m(y x0), -[infinity] < y < [infinity], the conditional distribution function of Y given X = x0, -[infinity] < x0 < [infinity]. 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([lambda] x0), 0 < [lambda] < 1, of m-1([lambda] 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.