Convergence rates of monotone conditional quantile estimators

Let {(Xi,Yi): 1 [less-than-or-equals, slant] i[less-than-or-equals, slant]n} be a sample of size n from a bivariate population, 0 < p < 1, and let [xi]p(x) be the p-quantile of Y1 given X1 = x. Estimation of [xi]p(x) when it is monotone in x for all 0 < p < 1, has been studied in the literature. Casady and Cryer (1976) constructed a monotone estimator [xi]*pn(x) and showed that the a.s. convergence rate for the estimator is O((log log n/n)1/4). Wright (1984) showed that n1/3([xi]*pn(x) - [xi]p(x)) has a non-degenerate limit distribution for all x and 0 < p < 1. We show that [xi]*pn(x) - [xi]p(x) = O((log n/n)1/3) a.s. for all x and 0 < p < 1.