A Kernel Estimator of a Conditional Quantile

Let (X1, Y1), (X2, Y2), …, be two-dimensional random vectors which are independent and distributed as (X, Y). For 0<p<1, letξ(p∣x) be the conditionalpth quantile ofYgivenX=x; that is,ξ(p∣x)=inf{y : P(Y⩽y∣X=x)⩾p}. We consider the problem of estimatingξ(p∣x) from the data (X1, Y1), (X2, Y2), …, (Xn, Yn). In this paper, a new kernel estimator ofξ(p∣x) is proposed. The asymptotic normality and a law of the iterated logarithm are obtained.