Optimal Global Rates of Convergence for Nonparametric Regression

Consider a p-times differentiable unknown regression function of a d-dimensional measurement variable. Let T () denote a derivative of of order m and set r = (p - m) / (2p + d). Let Tₙ denote an estimator of T () based on a training sample of size n, and let \| Tₙ - T () \|q be the usual Lq norm of the restriction of Tₙ - T () to a fixed compact set. Under appropriate regularity conditions, it is shown that the optimal rate of convergence for \| Tₙ - T () \|q is n^-r if 0 < q < ; while (n^-1 n) ʳ is the optimal rate if q =.