A kernel estimator of a conditional quantile

Let (X(1), Y-1), (X(2), Y-2), ..., be two-dimensional random vectors which are independent and distributed as (X, Y). For 0 < p < 1, let xi((p\x) be the conditional pth quantile of Y given X = x; that is, xi(p\x) = inf{y:P(Y less than or equal to y\X = x) greater than or equal to p}. We consider the problem of estimating xi(p\x) from the data (X(1), Y-1), (X(2), Y-2), ..., (X(n), Y-n). In this paper, a new kernel estimator of xi(p\x) is proposed. The asymptotic normality and a law of the iterated logarithm are obtained. (C) 1996 Academic Press, Inc.