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

ISBN-10: 0387954414

ISBN-13: 9780387954417

This booklet presents a scientific in-depth research of nonparametric regression with random layout. It covers just about all recognized 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)**

**Sample text**

Yn } \ {Yi1 , . . , Yil } depends only on C \ {Cj,k } and on Xr ’s and Nr ’s with r ∈ {i1 , . . , il }, and therefore is independent of Cj,k given X1 , . . , Xn . Now conditioning on X1 , . . , Xn , the error of the conditional Bayes decision for Cj,k based on (Y1 , . . , Yn ) depends only on (Yi1 , . . 2 implies ⎛ ⎞ P{C¯n,j,k = Cj,k |X1 , . . , Xn } = Φ ⎝− l r=1 2 (X )⎠ gj,k ir 48 3. Lower Bounds ⎛ n Φ ⎝− = ⎞ 2 (X )⎠ . gj,k i i=1 √ Since Φ(− x) is convex, by Jensen’s inequality P{C¯n,j,k = Cj,k } = E{P{C¯n,j,k = Cj,k |X1 , .

N ∈ R+ , and some independent and identically distributed random variables 1 , . . , n with E 1 = 0 and E 21 = 1, one wants to estimate the values of f at the socalled design points x1 , . . , xn . Typically, in this problem, one has d = 1, sometimes also d = 2 (image reconstruction). , σ12 = · · · = σn2 = σ 2 . Clearly, this problem has some similarity with the problem we study in this book. 14) where i = i (Xi ) = Yi − m(Xi ) satisﬁes E{ i |Xi } = 0. It may seem that ﬁxed design regression is a more general approach than random design and that one can handle random design regression estimation by imposing conditions on the design points and then applying 16 1.

13) for some (unknown) function f : Rd → R, some σ1 , . . , σn ∈ R+ , and some independent and identically distributed random variables 1 , . . , n with E 1 = 0 and E 21 = 1, one wants to estimate the values of f at the socalled design points x1 , . . , xn . Typically, in this problem, one has d = 1, sometimes also d = 2 (image reconstruction). , σ12 = · · · = σn2 = σ 2 . Clearly, this problem has some similarity with the problem we study in this book. 14) where i = i (Xi ) = Yi − m(Xi ) satisﬁes E{ i |Xi } = 0.

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

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