Asymptotic Properties of Kernel Estimators Based on Local Medians

The desire to make nonparametric regression robust leads to the problem of conditional median function estimation. Under appropriate regularity conditions, a sequence of local median estimators can be chosen to achieve the optimal rate of convergence n^-1/ (2+d) both pointwise and in the Lq (1 q <) norm restricted to a compact. It can also be chosen to achieve the optimal rate of convergence (n^-1 n) ^1/ (2+d) in the L^ norm restricted to a compact. These results also constitute an answer to an open question of Stone.