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 rn(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[rn(X) − r(X)]2, and particularize our results to classical function spaces such as Sobolev spaces, Besov spaces and reproducing kernel Hilbert spaces. Index Terms — Regression estimation, Nearest neighbor estimate, Rates of convergence, Compact embedding, Reproducing kernel Hilbert space, Sobolev space. AMS 2000 Classification: 62G05, 62G08. Corresponding author.