Asymptotic Theory of Statistics and Probability

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Springer Science & Business Media, Mar 7, 2008 - Mathematics - 722 pages
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This book is an encyclopedic treatment of classic as well as contemporary large sample theory, dealing with both statistical problems and probabilistic issues and tools. It is written in an extremely lucid style, with an emphasis on the conceptual discussion of the importance of a problem and the impact and relevance of the theorems. The book has 34 chapters over a wide range of topics, nearly 600 exercises for practice and instruction, and another 300 worked out examples. It also includes a large compendium of 300 useful inequalities on probability, linear algebra, and analysis that are collected together from numerous sources, as an invaluable reference for researchers in statistics, probability, and mathematics.

It can be used as a graduate text, as a versatile research reference, as a source for independent reading on a wide assembly of topics, and as a window to learning the latest developments in contemporary topics. The book is unique in its detailed coverage of fundamental topics such as central limit theorems in numerous setups, likelihood based methods, goodness of fit, higher order asymptotics, as well as of the most modern topics such as the bootstrap, dependent data, Bayesian asymptotics, nonparametric density estimation, mixture models, and multiple testing and false discovery. It provides extensive bibliographic references on all topics that include very recent publications.

Anirban DasGupta is Professor of Statistics at Purdue University. He has also taught at the Wharton School of the University of Pennsylvania, at Cornell University, and at the University of California at San Diego. He has been on the editorial board of the Annals of Statistics since 1998 and has also served on the editorial boards of the Journal of the American Statistical Association, International Statistical Review, and the Journal of Statistical Planning and Inference. He has edited two monographs in the lecture notes monograph series of the Institute of Mathematical Statistics, is a Fellow of the Institute of Mathematical Statistics and has 70 refereed publications on theoretical statistics and probability in major journals.

  

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Contents

Basic Convergence Concepts and Theorems
1
12 Three Series Theorem and Kolmogorovs ZeroOne Law
6
13 Central Limit Theorem and Law of the Iterated Logarithm
7
14 Further Illustrative Examples
10
15 Exercises
12
References
16
Metrics Information Theory Convergence and Poisson Approximations
19
21 Some Common Metrics and Their Usefulness
20
215 Bartlett Correction
338
216 The Wald and Rao Score Tests
339
217 Likelihood Ratio Confidence Intervals
340
218 Exercises
342
References
344
Asymptotic Efficiency in Testing
347
221 Pitman Efficiencies
348
222 Bahadur Slopes and Bahadur Efficiency
353

22 Convergence in Total Variation and Further Useful Formulas
22
23 InformationTheoretic Distances de Bruijns Identity and Relations to Convergence
24
24 Poisson Approximations
28
25 Exercises
31
References
33
More General Weak and Strong Laws and the Delta Theorem
35
32 Median Centering and Kestens Theorem
38
33 The Ergodic Theorem
39
34 Delta Theorem and Examples
40
35 Approximation of Moments
44
36 Exercises
45
References
47
Transformations
49
41 VarianceStabilizing Transformations
50
42 Examples
51
43 Bias Correction of the VST
54
44 Symmetrizing Transformations
57
45 VST or Symmetrizing Transform?
59
References
61
More General Central Limit Theorems
63
52 CLT without a Variance
66
53 Combinatorial CLT
67
54 CLT for Exchangeable Sequences
68
55 CLT for a Random Number of Summands
70
56 Infinite Divisibility and Stable Laws
71
57 Exercises
77
References
80
Moment Convergence and Uniform Integrability
83
62 The Moment Problem
85
63 Exercises
88
Referencesq
89
Sample Percentiles and Order Statistics
91
71 Asymptotic Distribution of One Order Statistic
92
72 Joint Asymptotic Distribution of Several Order Statistics
93
73 Bahadur Representations
94
74 Confidence Intervals for Quantiles
96
75 Regression Quantiles
97
76 Exercises
98
References
100
Sample Extremes
101
82 Characterizations
105
83 Limiting Distribution of the Sample Range
107
84 Multiplicative Strong Law
108
85 Additive Strong Law
109
86 Dependent Sequences
111
Sample Extremes
114
References
116
Central Limit Theorems for Dependent Sequences
119
92 Sampling without Replacement
121
93 Martingales and Examples
123
94 The Martingale and Reverse Martingale CLTs
126
References
129
Central Limit Theorem for Markov Chains
131
102 Normal Limits
132
103 Nonnormal Limits
135
105 Exercises
137
References
139
Accuracy of Central Limit Theorems
141
BerryEsseen Inequality
142
112 Local Bounds
144
113 The Multidimensional BerryEsseen Theorems
145
114 Other Statistics
146
115 Exercises
147
References
149
Invariance Principles
151
121 Motivating Examples
152
122 Two Relevant Gaussian Processes
153
123 The ErdösKac Invariance Principle
156
124 Invariance Principles Donskers Theorem and the KMT Construction
157
125 Invariance Principle for Empirical Processes
161
126 Extensions of Donskers Principle and VapnikChervonenkis Classes
163
127 GlivenkoCantelli Theorem for VC Classes
164
128 CLTs for Empirical Measures and Applications
167
1281 Notation and Formulation
168
1282 Entropy Bounds and Specific CLTs
169
Martingales Mixing and ShortRange Dependence
172
1210 Weighted Empirical Processes and Approximations
175
1211 Exercises
178
References
180
Edgeworth Expansions and Cumulants
185
131 Expansion for Means
186
132 Using the Edgeworth Expansion
188
133 Edgeworth Expansion for Sample Percentiles
189
134 Edgeworth Expansion for the 𝒕statistic
190
135 CornishFisher Expansions
192
136 Cumulants and Fishers 𝜿statistics
194
137 Exercises
198
References
200
Saddlepoint Approximations
203
141 Approximate Evaluation of Integrals
204
142 Density of Means and Exponential Tilting
208
1421 Derivation by Edgeworth Expansion and Exponential Tilting
210
143 Some Examples
211
144 Application to Exponential Family and the Magic Formula
213
146 Edgeworth vs Saddlepoint vs Chisquare Approximation
217
147 Tail Areas for Sample Percentiles
218
148 Quantile Approximation and Inverting the LugannaniRice Formula
219
149 The Multidimensional Case
221
1410 Exercises
222
References
223
𝑼statistics
225
151 Examples
226
152 Asymptotic Distribution of 𝑼statistics
227
153 Moments of 𝑼statistics and the Martingale Structure
229
154 Edgeworth Expansions
230
155 Nonnormal Limits
232
References
234
Maximum Likelihood Estimates
235
162 Inconsistent MLEs
239
163 MLEs in the Exponential Family
240
164 More General Cases and Asymptotic Normality
242
165 Observed and Expected Fisher Information
244
166 Edgeworth Expansions for MLEs
245
167 Asymptotic Optimality of the MLE and Superefficiency
247
168 Ha𝓳ekLeCam Convolution Theorem
249
169 Loss of Information and Efrons Curvature
251
1610 Exercises
253
References
258
M Estimates
259
171 Examples
260
172 Consistency and Asymptotic Normality
262
173 Bahadur Expansion of M Estimates
265
174 Exercises
267
References
268
The Trimmed Mean
271
182 Lower Bounds on Efficiencies
273
184 The 10203040 Rule
275
185 Exercises
277
References
278
Multivariate Location Parameter and Multivariate Medians
279
192 Multivariate Medians
280
193 Asymptotic Theory for Multivariate Medians
282
194 The Asymptotic Covariance Matrix
283
195 Asymptotic Covariance Matrix of the L₁ Median
284
196 Exercises
287
References
288
Bayes Procedures and Posterior Distributions
289
201 Motivating Examples
290
202 Bernsteinvon Mises Theorem
291
203 Posterior Expansions
294
204 Expansions for Posterior Mean Variance and Percentiles
298
205 The TierneyKadane Approximations
300
206 Frequentist Approximation of Posterior Summaries
302
207 Consistency of Posteriors
304
208 The Difference between Bayes Estimates and the MLE
305
209 Using the Brown Identity to Obtain Bayesian Asymptotics
306
2010 Testing
311
2011 Interval and Set Estimation
312
2012 InfiniteDimensional Problems and the DiaconisFreedman Results
314
2013 Exercises
317
References
320
1 Testing Problems
323
212 Examples
324
213 Asymptotic Theory of Likelihood Ratio Test Statistics
334
214 Distribution under Alternatives
336
223 Bahadur Slopes of 𝑼statistics
361
224 Exercises
362
References
363
Some General LargeDeviation Results
365
232 The GärtnerEllis Theorem
367
233 Large Deviation for Local Limit Theorems
370
234 Exercises
374
References
375
Classical Nonparametrics
377
241 Some Early Illustrative Examples
378
242 Sign Test
380
243 Consistency of the Sign Test
381
244 Wilcoxon SignedRank Test
383
245 Robustness of the 𝑡 Confidence Interval
388
246 The BahadurSavage Theorem
393
247 KolmogorovSmirnov and Anderson Confidence Intervals
394
248 HodgesLehmann Confidence Interval
396
249 Power of the Wilcoxon Test
397
2410 Exercises
398
References
399
TwoSample Problems
401
251 BehrensFisher Problem
402
252 Wilcoxon Rank Sum and MannWhitney Test
405
253 TwoSample 𝑼statistics and Power Approximations
408
254 Hettmanspergers Generalization
410
255 The Nonparametric BehrensFisher Problem
412
256 Robustness of the MannWhitney Test
415
257 Exercises
417
References
418
Goodness of Fit
421
261 KolmogorovSmirnov and Other Tests Based on 𝐹𝑛
422
263 Some Heuristics
423
264 Asymptotic Null Distributions of D𝑛 𝐶𝑛 𝐴𝑛 and 𝑉𝑛
424
265 Consistency and Distributions under Alternatives
425
266 Finite Sample Distributions and Other EDFBased Tests
426
267 The BerkJones Procedure
428
268 𝜑Divergences and the JagerWellner Tests
429
269 The TwoSample Case
431
2610 Tests for Normality
434
2611 Exercises
436
References
438
Chisquare Tests for Goodness of Fit
441
272 Asymptotic Distribution of Pearsons Chisquare
442
274 Choice of 𝒌
443
275 Recommendation of Mann and Wald
445
277 Exercises
448
References
449
Goodness of Fit with Estimated Parameters
451
281 Preliminary Analysis by Stochastic Expansion
452
282 Asymptotic Distribution of EDFBased Statistics for Composite Nulls
453
283 Chisquare Tests with Estimated Parameters and the ChernoffLehmann Result
455
284 Chisquare Tests with Random Cells
457
References
458
The Bootstrap
461
291 Bootstrap Distribution and the Meaning of Consistency
462
292 Consistency in the Kolmogorov and Wasserstein Metrics
464
293 Delta Theorem for the Bootstrap
468
295 Other Statistics
471
296 Some Numerical Examples
473
297 Failure of the Bootstrap
475
298 𝑚 out of 𝑛 Bootstrap
476
299 Bootstrap Confidence Intervals
478
2910 Some Numerical Examples
482
2911 Bootstrap Confidence Intervals for Quantiles
483
2913 Residual Bootstrap
484
2914 Confidence Intervals 485
485
2915 Distribution Estimates in Regression
486
2916 Bootstrap for Dependent Data
487
2917 Consistent Bootstrap for Stationary Autoregression
488
2918 Block Bootstrap Methods
489
2919 Optimal Block Length
491
2920 Exercises
492
References
495
Jackknife
499
302 Bias Correction by the Jackknife
502
303 Variance Estimation
503
304 Delete𝑑 Jackknife and von Mises Functionals
504
305 A Numerical Example
507
306 Jackknife Histogram
508
307 Exercises
511
References
512
Permutation Tests
513
311 General Permutation Tests and Basic Group Theory
514
312 Exact Similarity of Permutation Tests
516
313 Power of Permutation Tests
519
314 Exercises
520
References
521
Density Estimation
523
322 Measures of the Quality of Density Estimates
526
324 Minimaxity Criterion
529
A Preview
530
326 Rate of Convergence of Histograms
531
327 Consistency of Kernel Estimates
533
328 Order of Optimal Bandwidth and Superkernels
535
329 The Epanechnikov Kernel
538
3210 Choice of Bandwidth by Cross Validation
539
32101 Maximum Likelihood CV
540
32102 Least Squares CV
542
32103 Stones Result
544
3211 Comparison of Bandwidth Selectors and Recommendations
545
3212 𝓛 Optimal Bandwidths
547
3213 Variable Bandwidths
548
3214 Strong Uniform Consistency and Confidence Bands
550
3215 Multivariate Density Estimation and Curse of Dimensionality
552
32151 Kernel Estimates and Optimal Bandwidths
556
3216 Estimating a Unimodal Density and the Grenander Estimate
558
3217 Mode Estimation and Chernoffs Distribution
561
3218 Exercises
564
References
568
Mixture Models and Nonparametric Deconvolution
571
331 Mixtures as Dense Families
572
332 𝑧 Distributions and Other Gaussian Mixtures as Useful Models
573
Finite Mixtures
577
3332 Minimum Distance Method
578
3333 Moment Estimates
579
334 Estimation in General Mixtures
580
335 Strong Consistency and Weak Convergence of the MLE
582
336 Convergence Rates for Finite Mixtures and Nonparametric Deconvolution
584
3361 Nonparametric Deconvolution
585
337 Exercises
587
References
589
HighDimensional Inference and False Discovery
593
341 Chisquare Tests with Many Cells and Sparse Multinomials
594
The Portnoy Paradigm
597
Early Developments
599
Definitions Control and the BenjaminiHochberg Rule
601
345 Distribution Theory for False Discoveries and Poisson and FirstPassage Asymptotics
604
346 Newer FDR Controlling Procedures
606
347 Higher Criticism and the DonohoJin Developments
608
348 False Nondiscovery and Decision Theory Formulation
611
3481 GenoveseWasserman Procedure
612
349 Asymptotic Expansions
614
3410 Lower Bounds on the Number of False Hypotheses
616
34101 BühlmannMeinshausenRice Method
617
3411 The Dependent Case and the HallJin Results
620
HallJin Results
623
3412 Exercises
625
References
628
A Collection of Inequalities in Probability Linear Algebra and Analysis
633
3512 Concentration Inequalities
634
3513 Tail Inequalities for Specific Distributions
639
3514 Inequalities under Unimodality
641
3515 Moment and Monotonicity Inequalities
643
3516 Inequalities in Order Statistics
652
3517 Inequalities for Normal Distributions
655
3518 Inequalities for Binomial and Poisson Distributions
656
3519 Inequalities in the Central Limit Theorem
658
35110 Martingale Inequalities
661
352 Matrix Inequalities
663
3522 Eigenvalue and Quadratic Form Inequalities
667
353 Series and Polynomial Inequalities
671
354 Integral and Derivative Inequalities
675
Glossary of Symbols
689
Index
693
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About the author (2008)

Anirban DasGupta is Professor of Statistics at Purdue University.

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