Rates of Convergence of the Functional $k$ -Nearest Neighbor Estimate

Let F be a separable Banach space, and let (X, Y) be a random pair taking values in F × R. Motivated by a broad range of potential applications, we investigate rates of convergence of the k-nearest neighbor estimate r_n (x) of the regression function r(x) = E[Y|X = x], based on n independent copies of the pair (X, Y). Using compact embedding theory, we present explicit and general finite sample bounds on the expected squared difference E[r_n(X) - r(X)]², and particularize our results to classical function spaces such as Sobolev spaces, Besov spaces, and reproducing kernel Hilbert spaces.