Download e-book for kindle: A Distribution-Free Theory of Nonparametric Regression by László Györfi, Michael Kohler, Adam Krzyzak, Harro Walk

By László Györfi, Michael Kohler, Adam Krzyzak, Harro Walk

ISBN-10: 0387954414

ISBN-13: 9780387954417

 This e-book presents a scientific in-depth research of nonparametric regression with random layout. It covers just about all identified estimates. The emphasis is on distribution-free homes of the estimates.

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Additional resources for A Distribution-Free Theory of Nonparametric Regression (Springer Series in Statistics)

Example text

Let D(p,C) be the class of distributions of (X, Y ) such that: (i) X is uniformly distributed on [0, 1]d ; (ii) Y = m(X)+N , where X and N are independent and N is standard normal; and (iii) m ∈ F (p,C) . In the next theorem we derive a lower minimax rate of convergence for this class of distributions. 2. For the class D(p,C) , the sequence 2p an = n− 2p+d is a lower minimax rate of convergence. In particular, lim inf inf sup n→∞ mn (X,Y )∈D (p,C) E{ mn − m 2 } 2d 2p C 2p+d n− 2p+d ≥ C1 > 0 for some constant C1 independent of C.

D ), αi ∈ N0 , and d j=1 αj = k. ∂xd d . If |Dα m(cn ) (x) − Dα m(cn ) (z)| = |cn,i | · |Dα gn,i (x) − Dα gn,i (z)| ≤ C2β−1 Mn−p Mnk Mn (x − an,i ) − Mn (z − an,i ) ≤ C2β−1 x − z ≤ C x−z β β β . Now assume that x ∈ An,i and z ∈ An,j for i = j. 2. Minimax Lower Bounds 41 boundary of An,j , and x − x ¯ + z¯ − z ≤ x − z . Then |Dα m(cn ) (x) − Dα m(cn ) (z)| = |cn,i Dα gn,i (x) − cn,j Dα gn,j (z)| ≤ |cn,i Dα gn,i (x)| + |cn,j Dα gn,j (z)| = |cn,i | · |Dα gn,i (x) − Dα gn,i (¯ x)| + |cn,j | · |Dα gn,j (z) − Dα gn,j (¯ z )| x) = Dα gn,j (¯ z ) = 0) (because of Dα gn,i (¯ ≤ C2β−1 ( x − x ¯ + z − z¯ β ) β (as in the first case) = C2β 1 x−x ¯ 2 ≤ C2β β + 1 z − z¯ 2 β β z − z¯ x−x ¯ + 2 2 (by Jensen’s inequality) ≤ C x−z β .

Galton’s 1889 work has been continued by F. Y. Edgeworth, K. Pearson, G. U. Yule, and R. A. Fisher in the last decade of the nineteenth century and in the first decade of the twentieth century. For historical details we refer to Hald (1998), Farebrother (1999), and Stigler (1999). The principle of penalized modeling, in particular, smoothing splines, goes back to Whittaker (1923), Schoenberg (1964), and Reinsch (1967); see Wahba (1990) for additional references. The phrase “curse of dimensionality” is due to Bellman (1961).

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A Distribution-Free Theory of Nonparametric Regression (Springer Series in Statistics) by László Györfi, Michael Kohler, Adam Krzyzak, Harro Walk

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