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, let[xi](p|x) be the conditionalpth quantile ofYgivenX=x; that is,[xi](p|x)=inf{y : P(Y[less-than-or-equals, slant]y|X=x)[greater-or-equal, slanted]p}. We consider the problem of estimating[xi](p|x) from the data (X1, Y1), (X2, Y2), ..., (Xn, Yn). 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.