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

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

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Example text

The most popular example of a local modeling estimate is the local polynomial kernel estimate. 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 .

In this monograph we are interested in properties of mn that are valid for all distributions of (X, Y ), that is, in distribution-free or universal properties. The concept of universal consistency is important in nonparametric regression because the mere use of a nonparametric estimate is normally a consequence of the partial or total lack of information about the distribution of (X, Y ). Since in many situations we do not have any prior information about the distribution, it is essential to have estimates that perform well for all distributions.

2. A sequence of regression function estimates {mn } is called strongly consistent for a certain distribution of (X, Y ), if lim n→∞ (mn (x) − m(x))2 µ(dx) = 0 with probability one. It may be that a regression function estimate is consistent for a certain class of distributions of (X, Y ), but not consistent for others. It is clearly desirable to have estimates that are consistent for a large class of distributions. In this monograph we are interested in properties of mn that are valid for all distributions of (X, Y ), that is, in distribution-free or universal properties.

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