Convergence rate of b-spline estimators of nonparametric conditional quantile functions

Given a bivariate sample {(Xi,Yi), i = 1,2,…, n}, we consider the problem of estimating the conditional quantile functions of nonparametric regression by minimizing ∑ρα(Yi-g(Xi)) over g in a linear space of B-spline functions, where ρα(u) = |u| - (2α - 1)u is the Czech function of Koenker and Bassett (1978). If the true conditional quantile function is smooth up to order r, we show that the optimal global convergence rate of n -r/(2r+1) is attained by the B-spline based estimators if the number of knots is in the order of n 1/(2r+1).