## K-Nearest neighbor estimation |

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absolutely continuous asymptotic behavior asymptotic bias asymptotic independence asymptotic normality bandwidth bias and variance Bickel and Rosenblatt bounded derivatives bounded variation Brownian bridge Brownian motion cf(x Chebychev's inequality concomitants consider continuously differentiable defined density f(x derive the asymptotic differentiable with bounded Euclidean distance extreme value distribution f is continuously fn(x Fukunaga and Hostetler Gaussian process gn(x goodness-of-fit problem i.i.d. sample joint density k-NN density estimate k-NN estimates k-NN method k-NN regression estimate kernel density estimates kernel estimate kernel method kth n n nearest kth nearest neighbor kth order statistic Lebesgue measure Lemma Loftsgaarden and Quesenberry marginal density mean square error Moore and Yackel n n nearest neighbor nearest neighbor estimate neighbor density estimate nP(SR obtain P(SR positive integers Professor Murray Rosenblatt Proof regression function rn(x sequence of positive Taylor expansions uniform consistency uniformly continuous univariate University of California weight function Yackel 33