Optimal Global Rates of Convergence for Nonparametric Regression

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