Optimal Rates of Convergence for Nonparametric Estimators

Let d denote a positive integer, \|x\| = (x²₁ + + x²d) ^1/2 the Euclidean norm of x = (x₁, , xd) Rᵈ, k a nonnegative integer, Cₖ the collection of k times continuously differentiable functions on Rᵈ, and gₖ the Taylor polynomial of degree k about the origin corresponding to g Cₖ. Let M and p > k denote positive constants and let U be an open neighborhood of the origin of Rᵈ. Let G denote the collection of functions g Cₖ such that |g (x) - gₖ (x) | M \|x\|P for x U. Let m k be a nonnegative integer, let ₀ₘ and set = \₀ + g: g G\. Let L be a linear differential operator of order m on Cₘ and set T () = L (0) for. Let (X, Y) be a pair of random variables such that X is Rᵈ valued and Y is real valued. It is assumed that the distribution of X is absolutely continuous and that its density is bounded away from zero and infinity on U. The conditional distribution of Y given X is assumed to be (say) normal, with a conditional variance which is bounded away from zero and infinity on U. The regression function of Y on X is assumed to belong to. It is shown that r = (p - m) / (2p + d) is the optimal (uniform) rate of convergence for a sequence \Tₙ\ of estimators of T () such that Tₙ is based on a random sample of size n from the distribution of (X, Y). An analogous result is obtained for nonparametric estimators of a density function.