Download PDF by László Györfi, Michael Kohler, Adam Krzyzak, Harro Walk: A Distribution-free Theory of Nonparametric Regression

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

ISBN-10: 1441929983

ISBN-13: 9781441929983

This e-book offers a scientific in-depth research of nonparametric regression with random layout. It covers just about all recognized estimates akin to classical neighborhood averaging estimates together with kernel, partitioning and nearest neighbor estimates, least squares estimates utilizing splines, neural networks and radial foundation functionality networks, penalized least squares estimates, neighborhood polynomial kernel estimates, and orthogonal sequence estimates. The emphasis is on distribution-free homes of the estimates. such a lot consistency effects are legitimate for all distributions of the information. at any time when it's not attainable to derive distribution-free effects, as relating to the charges of convergence, the emphasis is on effects which require as few constrains on distributions as attainable, on distribution-free inequalities, and on adaptation.
The correct mathematical concept is systematically built and calls for just a uncomplicated wisdom of likelihood concept. The ebook may be a useful reference for a person attracted to nonparametric regression and is a wealthy resource of many beneficial mathematical recommendations largely scattered within the literature. particularly, the e-book introduces the reader to empirical technique conception, martingales and approximation houses of neural networks.

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Extra resources for A Distribution-free Theory of Nonparametric Regression

Example text

In the classical minimax approach one tries to minimize the maximal error within a class of distributions. If we use E mn − m 2 as error, then this means that one tries to minimize sup E{(mn (X) − m(X))2 }. (X,Y )∈D In the sequel we will derive asymptotic lower bounds of inf sup mn (X,Y )∈D E{(mn (X) − m(X))2 } for special classes D of distributions. , over all measurable functions of the data. 1. The sequence of positive numbers an is called the lower minimax rate of convergence for the class D if lim inf inf sup n→∞ mn (X,Y )∈D E{ mn − m 2 } = C1 > 0.

In other words, the “bad” distribution, causing the largest error for an estimator, may be different for each n. For example, the lower bound for the class D(p,C) does not exclude the possibility that there exists a sequence of estimators {mn } such that for every distribution in D(p,C) , the expected error E{ mn − m 2 } decreases at an exponential rate in n. In this section, we are interested in “individual” minimax lower bounds that describe the behavior of the error for a fixed distribution of (X, Y ) as the sample size n grows.

For parametric methods we refer to Rao (1973), Seber (1977), Draper and Smith (1981) and Farebrother (1988) and the literature cited therein. 1 are well-known in the literature. , Van Ryzin (1966), Wolverton and Wagner (1969a), Glick (1973), Csibi (1971), Gy¨orfi (1975; 1978), Devroye and Wagner (1976), Devroye (1982b), or Devroye and Gy¨orfi (1985). The concept of (weak) universal consistency goes back to Stone (1977). 1. Show that the regression function also has the following pointwise optimality property: E |m(X) − Y |2 |X = x = min E |f (X) − Y |2 |X = x f Problems and Exercises 17 for µ-almost all x ∈ Rd .

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A Distribution-free Theory of Nonparametric Regression by László Györfi, Michael Kohler, Adam Krzyzak, Harro Walk


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