A Distribution-Free Theory of Nonparametric Regression

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Springer Science & Business Media, Aug 12, 2002 - Mathematics - 647 pages
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This book provides a systematic in-depth analysis of nonparametric regression with random design. It covers almost all known estimates such as classical local averaging estimates including kernel, partitioning and nearest neighbor estimates, least squares estimates using splines, neural networks and radial basis function networks, penalized least squares estimates, local polynomial kernel estimates, and orthogonal series estimates. The emphasis is on distribution-free properties of the estimates. Most consistency results are valid for all distributions of the data. Whenever it is not possible to derive distribution-free results, as in the case of the rates of convergence, the emphasis is on results which require as few constrains on distributions as possible, on distribution-free inequalities, and on adaptation. The relevant mathematical theory is systematically developed and requires only a basic knowledge of probability theory. The book will be a valuable reference for anyone interested in nonparametric regression and is a rich source of many useful mathematical techniques widely scattered in the literature. In particular, the book introduces the reader to empirical process theory, martingales and approximation properties of neural networks.
 

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Contents

mparametric Regression Important?
1
stric versus Nonparametric Estimation
9
f Convergence 277
13
ersus Random Design Regression
15
A Dimensionality
23
inds
31
ig Estimates
52
ency
60
raphic Notes
281
f Convergence
294
sis Function Networks
329
Series Estimates
353
ency
366
Techniques from Empirical Process Theory
380
ise Polynomial Partitioning Estimates
397
iate Penalized Least Squares Estimates
408

raphic Notes
67
f Convergence
77
f Convergence
93
he Sample
100
raphic Notes
108
f Theorem 8 1
115
t Neighbor Estimates
126
Exponential Inequalities
131
ig and Packing Numbers
140
orm Law of Large Numbers
153
ency from Bounded to Unbounded Y
165
Rate of Convergence
183
Complexity Regularization
222
y of DataDependent Partitioning Estimates
235
Partitions with DataDependent Grid Size
241
raphic Notes
250
f Lemma 20 1
414
striate Penalized Least Squares Estimates
425
f Convergence
433
ition of Complexity Regularization
440
raphic notes
446
Index Models
456
sive Estimates
493
cursive Partitioning Estimate
507
ive Kernel Estimate
517
Dbservations
540
sion Estimation for Model B
555
Autoregression
569
ting Smooth Regression Functions
582
lities for Martingales
598
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