Summary: Let (X,Y) be a random vector such that X is d-dimensional, Y is real valued and , where X and are independent and the th quantile of is 0 ( is fixed such that . Assume that is a smooth function with order of smoothness , and set
where m is a nonnegative integer smaller than p. Let T() denote a derivative of of order m.
It is proved that there exists a pointwise estimate of T(), based on a set of i.i.d. observations , that achieves the optimal nonparametric rate of convergence under appropriate regularity conditions. Further, a local Bahadur type representation is shown to hold for the estimate and this is used to obtain some useful asymptotic results.