Weak Convergence and Empirical Processes: With Applications to Statistics

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Springer Science & Business Media, Mar 14, 1996 - Mathematics - 508 pages
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This book tries to do three things. The first goal is to give an exposition of certain modes of stochastic convergence, in particular convergence in distribution. The classical theory of this subject was developed mostly in the 1950s and is well summarized in Billingsley (1968). During the last 15 years, the need for a more general theory allowing random elements that are not Borel measurable has become well established, particularly in developing the theory of empirical processes. Part 1 of the book, Stochastic Convergence, gives an exposition of such a theory following the ideas of J. Hoffmann-J!1Jrgensen and R. M. Dudley. A second goal is to use the weak convergence theory background devel oped in Part 1 to present an account of major components of the modern theory of empirical processes indexed by classes of sets and functions. The weak convergence theory developed in Part 1 is important for this, simply because the empirical processes studied in Part 2, Empirical Processes, are naturally viewed as taking values in nonseparable Banach spaces, even in the most elementary cases, and are typically not Borel measurable. Much of the theory presented in Part 2 has previously been scattered in the journal literature and has, as a result, been accessible only to a relatively small number of specialists. In view of the importance of this theory for statis tics, we hope that the presentation given here will make this theory more accessible to statisticians as well as to probabilists interested in statistical applications.
 

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Contents

Introduction
2
Outer Integrals and Measurable Majorants
6
Weak Convergence
16
Product Spaces
29
Spaces of Bounded Functions
34
Spaces of Locally Bounded Functions
43
The Ball SigmaField and Measurability of Suprema
45
Hilbert Spaces
49
2141 Finite Entropy Integrals
239
2142 Uniformly Bounded Classes
245
2143 Deviations from the Mean
254
2144 Proof of Theorem 21413
257
Notes
269
Statistical Applications
277
Introduction
278
MEstimators
284

Convergence Almost Surely and in Probability
52
Convergence Weak Almost Uniform and in Probability
57
Refinements
67
Uniformity and Metrization
71
Notes
75
Empirical Processes
79
Introduction
80
211 Overview of Chapters 23214
83
212 Asymptotic Equicontinuity
89
213 Maximal Inequalities
90
214 The Central Limit Theorem in Banach Spaces
91
Maximal Inequalities and Covering Numbers
95
221 SubGaussian Inequalities
100
222 Bernsteins Inequality
102
223 Tightness Under an Increment Bound
104
Symmetrization and Measurability
107
232 More Symmetrization
111
233 Separable Versions
115
GlivenkoCantelli Theorems
122
Donsker Theorems
127
252 Bracketing
129
Uniform Entropy Numbers
134
262 VCClasses of Functions
141
263 Convex Hulls and VCHull Classes
142
264 VCMajor Classes
145
265 Examples and Permanence Properties
146
Bracketing Numbers
154
272 Monotone Functions
159
273 Closed Convex Sets and Convex Functions
162
274 Classes That Are Lipschitz in a Parameter
164
Uniformity in the Underlying Distribution
166
282 Donsker Theorems
168
283 Central Limit Theorem Under Sequences
173
Multiplier Central Limit Theorems
176
Permanence of the Donsker Property
190
2102 Lipschitz Transformations
192
2103 Permanence of the Uniform Entropy Bound
198
2104 Partitions of the Sample Space
200
The Central Limit Theorem for Processes
205
2112 Bracketing
210
2113 Classes of Functions Changing with n
220
PartialSum Processes
225
2122 PartialSum Processes on Lattices
228
Other Donsker Classes
232
2132 Elliptical Classes
233
2133 Classes of Sets
236
Tail Bounds
238
321 The Argmax Theorem
285
322 Rate of Convergence
289
323 Examples
294
324 Linearization
300
ZEstimators
309
Rates of Convergence
321
341 Maximum Likelihood
326
342 Concave Parametrizations
330
343 Least Squares Regression
331
344 LeastAbsoluteDeviation Regression
336
Random Sample Size Poissonization and Kac Processes
339
352 Poissonization
341
The Bootstrap
345
362 The Exchangeable Bootstrap
353
The TwoSample Problem
360
371 Permutation Empirical Processes
362
372 TwoSample Bootstrap
365
Independence Empirical Processes
367
The DeltaMethod
372
392 Gaussian Limits
376
393 The DeltaMethod for the Bootstrap
377
394 Examples of the DeltaMethod
381
310 Contiguity
401
3101 The Empirical Process
406
3102 ChangePoint Alternatives
408
Convolution and Minimax Theorems
412
3111 Efficiency of the Empirical Distribution
420
Notes
423
Appendix
429
Inequalities
430
Gaussian Processes
437
A22 Exponential Bounds
442
A23 Majorizing Measures
445
A24 Further Results
447
Rademacher Processes
449
Isoperimetric Inequalities for Product Measures
451
Some Limit Theorems
456
More Inequalities
459
A62 Multinomial Random Vectors
462
A63 Rademacher Sums
463
Notes
465
References
467
Author Index
487
Subject Index
493
List of Symbols
506
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