New PDF release: A Distribution-Free Theory of Nonparametric Regression

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

ISBN-10: 0387954414

ISBN-13: 9780387954417

 This ebook offers a scientific in-depth research of nonparametric regression with random layout. It covers just about all recognized estimates. The emphasis is on distribution-free houses of the estimates.

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Extra info for A Distribution-Free Theory of Nonparametric Regression (Springer Series in Statistics)

Example text

Here one chooses a finite or countably infinite partition Pn = {An,1 , An,2 , . 1) where IA denotes the indicator function of set A, so Wn,i (x) = I{Xi ∈An,j } n l=1 I{Xl ∈An,j } for x ∈ An,j . Here and in the following we use the convention 00 = 0. The second example of a local averaging estimate is the Nadaraya– Watson kernel estimate. Let K : Rd → R+ be a function called the kernel function, and let h > 0 be a bandwidth. 2) so Wn,i (x) = x−Xi h x−Xj n K j=1 h K . , one estimates m(x) by averaging Yi ’s such that the distance between Xi and x is not greater than h.

Here one locally fits a polynomial to the data. For example, for d = 1, X is real-valued and l l g x, {ak }k=1 = ak xk−1 k=1 is a polynomial of degree l − 1 (or less) in x. A generalization of the partitioning estimate leads to global modeling or least squares estimates. Let Pn = {An,1 , An,2 , . , ⎧ ⎫ ⎨ ⎬ Fn = aj IAn,j : aj ∈ R . 7) ⎩ ⎭ j Then it is easy to see (cf. 1) satisfies mn (·) = arg min f ∈Fn 1 n n |f (Xi ) − Yi |2 . 9) i=1 over Fn . 7)). 9) over all (measurable) functions f , because this may lead to a function which interpolates the data and hence is not a reasonable estimate.

Xα . . ∂xd d β d (x, z ∈ Rd ). Let F (p,C) be the set of all (p, C)-smooth functions f : Rd → R. ¯ and {an } is an lower minimax rate of convergence Clearly, if D ⊆ D ¯ Thus, to for D, then it is also a lower minimax rate of convergence for D. determine lower minimax rates of convergence, it might be useful to restrict the class of distributions. It turns out that it suffices to look at classes of distributions where X is uniformly distributed on [0, 1]d and Y − m(X) has a normal distribution.

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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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