Normal approximation by Stein's method:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
2011
|
| Series: | Probability and its applications
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XII, 405 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9783642150067 |
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MARC
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Record in the Search Index
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IMAGE 1
CONTENTS
1 INTRODUCTION 1
1.1 THE CENTRAL LIMIT THEOREM 1
1.2 A BRIEF HISTORY OF STEIN'S METHOD 3
1.3 THE BASIC IDEA OF STEIN'S METHOD 4
1.4 OUTLINE AND SUMMARY 8
2 FUNDAMENTALS OF STEIN'S METHOD 13
2.1 STEIN'S EQUATION 13
2.2 PROPERTIES OF THE SOLUTIONS 15
2.3 CONSTRUCTION OF STEIN IDENTITIES 18
2.3.1 SUMS OF INDEPENDENT RANDOM VARIABLES 19
2.3.2 EXCHANGEABLE PAIRS 21
2.3.3 ZERO BIAS 26
2.3.4 SIZE BIAS 31
2.4 A GENERAL FRAMEWORK FOR STEIN IDENTITIES AND NORMAL APPROXIMATION
FOR LIPSCHITZ FUNCTIONS 36
APPENDIX 37
3 BERRY-ESSEEN BOUNDS FOR INDEPENDENT RANDOM VARIABLES 45 3.1 NORMAL
APPROXIMATION WITH LIPSCHITZ FUNCTIONS 46
3.2 THE LINDEBERG CENTRAL LIMIT THEOREM 48
3.3 BERRY-ESSEEN INEQUALITY: THE BOUNDED CASE 49
3.4 THE BERRY-ESSEEN INEQUALITY FOR UNBOUNDED VARIABLES 53 3.4.1 THE
CONCENTRATION INEQUALITY APPROACH 53
3.4.2 AN INDUCTIVE APPROACH 57
3.5 A LOWER BERRY-ESSEEN BOUND 59
4 L L BOUNDS 63
4.1 SUMS OF INDEPENDENT VARIABLES 65
4.1.1 L 1 BERRY-ESSEEN BOUNDS 65
4.1.2 CONTRACTION PRINCIPLE 69
IX
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1004244177
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
4.2 HIERARCHICAL STRUCTURES 72
4.2.1 BOUNDS TO THE NORMAL FOR APPROXIMATELY LINEAR RECURSIONS 78 4.2.2
NORMAL BOUNDS FOR HIERARCHICAL SEQUENCES 82
4.2.3 CONVERGENCE RATES FOR THE DIAMOND LATTICE 87
4.3 CONE MEASURE PROJECTIONS 88
4.3.1 COUPLING CONSTRUCTIONS FOR COORDINATE SYMMETRIC VARIABLES AND
THEIR PROJECTIONS 90
4.3.2 CONSTRUCTION AND BOUNDS FOR CONE MEASURE 94
4.4 COMBINATORIAL CENTRAL LIMIT THEOREMS 100
4.4.1 USE OF THE EXCHANGEABLE PAIR 102
4.4.2 CONSTRUCTION AND BOUNDS FOR THE COMBINATORIAL CENTRAL LIMIT
THEOREM 105
4.5 SIMPLE RANDOM SAMPLING I LL
4.6 CHATTERJEE'S L X THEOREM 116
4.7 LOCALLY DEPENDENT RANDOM VARIABLES 133
4.8 SMOOTH FUNCTION BOUNDS 136
4.8.1 FAST RATES FOR SMOOTH FUNCTIONS 136
APPENDIX 138
L BY BOUNDED COUPLINGS 147
5.1 BOUNDED ZERO BIAS COUPLINGS 147
5.2 EXCHANGEABLE PAIRS, KOLMOGOROV DISTANCE 149
5.3 SIZE BIASING, KOLMOGOROV BOUNDS 156
5.4 SIZE BIASING AND SMOOTHING INEQUALITIES 161
L: APPLICATIONS 167
6.1 COMBINATORIAL CENTRAL LIMIT THEOREM 167
6.1.1 UNIFORM DISTRIBUTION ON THE SYMMETRIC GROUP 168 6.1.2 DISTRIBUTION
CONSTANT ON CONJUGACY CLASSES 183 6.1.3 DOUBLY INDEXED PERMUTATION
STATISTICS 201
6.2 PATTERNS IN GRAPHS AND PERMUTATIONS 202
6.3 THE LIGHTBULB PROCESS 210
6.4 ANTI-VOTER MODEL 213
6.5 BINARY EXPANSION OF A RANDOM INTEGER 217
DISCRETIZED NORMAL APPROXIMATION 221
7.1 POISSON BINOMIAL 222
7.2 SUM OF INDEPENDENT INTEGER VALUED RANDOM VARIABLES 227
NON-UNIFORM BOUNDS FOR INDEPENDENT RANDOM VARIABLES 233 8.1 A
NON-UNIFORM CONCENTRATION INEQUALITY 233
8.2 NON-UNIFORM BERRY-ESSEEN BOUNDS 237
UNIFORM AND NON-UNIFORM BOUNDS UNDER LOCAL DEPENDENCE . . . 245 9.1
UNIFORM AND NON-UNIFORM BERRY-ESSEEN BOUNDS 246
9.2 OUTLINE OF PROOFS 247
9.3 APPLICATIONS 253
IMAGE 3
CONTENTS XI
10 UNIFORM AND NON-UNIFORM BOUNDS FOR NON-LINEAR STATISTICS 257 10.1
INTRODUCTION AND MAIN RESULTS 257
10.2 APPLICATIONS 260
10.2.1 {/-STATISTICS 260
10.2.2 MULTI-SAMPLE {/-STATISTICS 262
10.2.3 L-STATISTICS 265
10.2.4 RANDOM SUMS OF INDEPENDENT RANDOM VARIABLES WITH NON-RANDOM
CENTERING 270
10.2.5 FUNCTIONS OF NON-LINEAR STATISTICS 273
10.3 UNIFORM AND NON-UNIFORM RANDOMIZED CONCENTRATION INEQUALITIES 277
APPENDIX 284
11 MODERATE DEVIATIONS 293
11.1 A CRAMER TYPE MODERATE DEVIATION THEOREM 293
11.2 APPLICATIONS 295
11.3 PRELIMINARY LEMMAS 298
11.4 PROOFS OF MAIN RESULTS 302
APPENDIX 310
12 MULTIVARIATE NORMAL APPROXIMATION 313
12.1 MULTIVARIATE NORMAL APPROXIMATION VIA SIZE BIAS COUPLINGS . . . 314
12.2 DEGREES OF RANDOM GRAPHS 315
12.3 MULTIVARIATE EXCHANGEABLE PAIRS 325
12.4 LOCAL DEPENDENCE, AND BOUNDS IN KOLMOGOROV DISTANCE 331
13 NON-NORMAL APPROXIMATION 343
13.1 STEIN'S METHOD VIA THE DENSITY APPROACH 343
13.1.1 THE STEIN CHARACTERIZATION AND EQUATION 344
13.1.2 PROPERTIES OF THE STEIN SOLUTION 346
13.2 L 1 AND L BOUNDS VIA EXCHANGEABLE PAIRS 347
13.3 THE CURIE-WEISS MODEL 353
13.4 EXPONENTIAL APPROXIMATION 358
13.4.1 SPECTRUM OF THE BERNOULLI-LAPLACE MARKOV CHAIN 358 13.4.2 FIRST
PASSAGE TIMES 363
APPENDIX 367
14 GROUP CHARACTERS AND MALIIAVIN CALCULUS 371
14.1 NORMAL APPROXIMATION FOR GROUP CHARACTERS 371
14.1.1 O(2N,R) 379
14.1.2 SO(2N+L,R) 380
14.1.3 USP(2N,C) 381
14.2 STEIN'S METHOD AND MALIIAVIN CALCULUS 381
APPENDIX 389
NOTATION 389
REFERENCES 391
IMAGE 4
XII CONTENTS
AUTHOR INDEX 399
SUBJECT INDEX 403 |
| any_adam_object | 1 |
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| author_facet | Chen, Louis H. Y. 1940- Goldstein, Larry Joel 1944- Shao, Qi-Man |
| author_role | aut aut aut |
| author_sort | Chen, Louis H. Y. 1940- |
| author_variant | l h y c lhy lhyc l j g lj ljg q m s qms |
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| dewey-full | 519.24 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.24 |
| dewey-search | 519.24 |
| dewey-sort | 3519.24 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| spelling | Chen, Louis H. Y. 1940- Verfasser (DE-588)112808506 aut Normal approximation by Stein's method Louis H. Y. Chen ; Larry Goldstein ; Qi-Man Shao Berlin ; Heidelberg Springer 2011 XII, 405 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Probability and its applications Literaturangaben Approximation (DE-588)4002498-2 gnd rswk-swf Stein-Schätzung (DE-588)7570767-6 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 s Approximation (DE-588)4002498-2 s Stein-Schätzung (DE-588)7570767-6 s DE-604 Goldstein, Larry Joel 1944- Verfasser (DE-588)142903345 aut Shao, Qi-Man Verfasser (DE-588)137328605 aut Erscheint auch als Online-Ausgabe 978-3-642-15007-4 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3500781&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020811988&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chen, Louis H. Y. 1940- Goldstein, Larry Joel 1944- Shao, Qi-Man Normal approximation by Stein's method Approximation (DE-588)4002498-2 gnd Stein-Schätzung (DE-588)7570767-6 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd |
| subject_GND | (DE-588)4002498-2 (DE-588)7570767-6 (DE-588)4121894-2 |
| title | Normal approximation by Stein's method |
| title_auth | Normal approximation by Stein's method |
| title_exact_search | Normal approximation by Stein's method |
| title_full | Normal approximation by Stein's method Louis H. Y. Chen ; Larry Goldstein ; Qi-Man Shao |
| title_fullStr | Normal approximation by Stein's method Louis H. Y. Chen ; Larry Goldstein ; Qi-Man Shao |
| title_full_unstemmed | Normal approximation by Stein's method Louis H. Y. Chen ; Larry Goldstein ; Qi-Man Shao |
| title_short | Normal approximation by Stein's method |
| title_sort | normal approximation by stein s method |
| topic | Approximation (DE-588)4002498-2 gnd Stein-Schätzung (DE-588)7570767-6 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd |
| topic_facet | Approximation Stein-Schätzung Wahrscheinlichkeitsverteilung |
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