Self-normalized processes: limit theory and statistical applications
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2009
|
| Series: | Probability and its applications
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 275 S. |
| ISBN: | 9783540856351 |
Staff View
MARC
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| 245 | 1 | 0 | |a Self-normalized processes |b limit theory and statistical applications |c Victor H. de la Peña ; Tze Leung Lai ; Qi-Man Shao |
| 246 | 1 | |a Self normalized processes | |
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Record in the Search Index
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| adam_text | CONTENTS
1 INTRODUCTION 1
PART I INDEPENDENT RANDOM VARIABLES
2 CLASSICAL LIMIT THEOREMS, INEQUALITIES AND OTHER TOOLS
7
2.1 CLASSICAL LIMIT THEOREMS 7
2.1.1 THE WEAK LAW, STRONG LAW AND LAW OF THE ITERATED
LOGARITHM 8
2.1.2 THE CENTRAL LIMIT THEOREM 9
2.1.3 CRAMER S MODERATE DEVIATION THEOREM 11
2.2 EXPONENTIAL INEQUALITIES FOR SAMPLE SUMS 11
2.2.1 SELF-NORMALIZED SUMS 11
2.2.2 TAIL PROBABILITIES FOR PARTIAL SUMS 13
2.3 CHARACTERISTIC FUNCTIONS AND EXPANSIONS RELATED TO THE CLT 17
2.3.1 CONTINUITY THEOREM AND WEAK CONVERGENCE 18
2.3.2 SMOOTHING, LOCAL LIMIT THEOREMS AND EXPANSIONS 19
2.4 SUPPLEMENTARY RESULTS AND PROBLEMS 21
3 SELF-NORMALIZED LARGE DEVIATIONS
25
3.1 A CLASSICAL LARGE DEVIATION THEOREM FOR SAMPLE SUMS 25
3.2 A LARGE DEVIATION THEOREM FOR SELF-NORMALIZED SUMS 27
3.2.1 REPRESENTATION BY SUPREMUM OVER LINEAR FUNCTIONS
OF(S
N
,V
N
2
) 27
3.2.2 PROOF OF THEOREM 3.1 28
3.3 SUPPLEMENTARY RESULTS AND PROBLEMS 31
4 WEAK CONVERGENCE OF SELF-NORMALIZED SUMS
33
4.1 SELF-NORMALIZED CENTRAL LIMIT THEOREM 33
4.2 NON-NORMAL LIMITING DISTRIBUTIONS FOR SELF-NORMALIZED SUMS 37
4.3 SUPPLEMENTARY RESULTS AND PROBLEMS 38
BIBLIOGRAFISCHE INFORMATIONEN
HTTP://D-NB.INFO/989741796
DIGITALISIERT DURCH
CONTENTS
STEIN S METHOD AND SELF-NORMALIZED BERRY-ESSEEN INEQUALITY
41
5.1 STEIN S METHOD 41
5.1.1 THE STEIN EQUATION 41
5.1.2 STEIN S METHOD: ILLUSTRATION OF MAIN IDEAS 44
5.1.3 NORMAL APPROXIMATION FOR SMOOTH FUNCTIONS 46
5.2 CONCENTRATION INEQUALITY AND CLASSICAL BERRY-ESSEEN BOUND 49
5.3 A SELF-NORMALIZED BERRY-ESSEEN INEQUALITY 52
5.3.1 PROOF: OUTLINE OF MAIN IDEAS 53
5.3.2 PROOF: DETAILS 55
5.4 SUPPLEMENTARY RESULTS AND PROBLEMS 60
SELF-NORMALIZED MODERATE DEVIATIONS AND LAWS OF THE ITERATED
LOGARITHM
63
6.1 SELF-NORMALIZED MODERATE DEVIATIONS: NORMAL CASE 63
6.1.1 PROOF OF THE UPPER BOUND 64
6.1.2 PROOF OF THE LOWER BOUND 66
6.2 SELF-NORMALIZED MODERATE DEVIATIONS: STABLE CASE 69
6.2.1 PRELIMINARY LEMMAS 70
6.2.2 PROOF OF THEOREM 6.6 76
6.3 SELF-NORMALIZED LAWS OF THE ITERATED LOGARITHM 81
6.4 SUPPLEMENTARY RESULTS AND PROBLEMS 84
CRAMER-TYPE MODERATE DEVIATIONS FOR SELF-NORMALIZED SUMS
87
7.1 SELF-NORMALIZED CRAMER-TYPE MODERATE DEVIATIONS 87
7.2 PROOF OF THEOREMS 9
0
7.2.1 PROOF OF THEOREMS 7.2,7.4 AND COROLLARIES 9
0
7.2.2 PROOF OF THEOREM 7.1 91
7.2.3 PROOF OF PROPOSITIONS 94
7.3 APPLICATION TO SELF-NORMALIZED LJL 96
7.4 CRAMER-TYPE MODERATE DEVIATIONS FOR TWO-SAMPLE /-STATISTICS 104
7.5 SUPPLEMENTARY RESULTS AND PROBLEMS 106
SELF-NORMALIZED EMPIRICAL PROCESSES AND {/-STATISTICS
107
8.1 SELF-NORMALIZED EMPIRICAL PROCESSES 107
8.2 SELF-NORMALIZED {/-STATISTICS 108
8.2.1 THE HOEFFDING DECOMPOSITION AND CENTRAL
LIMIT THEOREM 109
8.2.2 SELF-NORMALIZED {/-STATISTICS AND BERRY-ESSEEN BOUNDS 109
8.2.3 MODERATE DEVIATIONS FOR SELF-NORMALIZED {/-STATISTICS 110
8.3 PROOFS OF THEOREMS 8.5 AND 8.6 IL
L
8.3.1 MAIN IDEAS OF THE PROOF IL
L
8.3.2 PROOF OF THEOREM 8.6 112
8.3.3 PROOF OF THEOREM 8.5 113
8.3.4 PROOF OF PROPOSITION 8.7 113
8.4 SUPPLEMENTARY RESULTS AND PROBLEMS 119
CONTENTS XI
PART II MARTINGALES AND DEPENDENT RANDOM VECTORS
9 MARTINGALE INEQUALITIES AND RELATED TOOLS
123
9.1 BASIC MARTINGALE THEORY 123
9.1.1 CONDITIONAL EXPECTATIONS AND MARTINGALES 123
9.1.2 MARTINGALE CONVERGENCE AND INEQUALITIES 125
9.2 TANGENT SEQUENCES AND DECOUPLING INEQUALITIES 125
9.2.1 CONSTRUCTION OF DECOUPLED TANGENT SEQUENCES 126
9.2.2 EXPONENTIAL DECOUPLING INEQUALITIES 126
9.3 EXPONENTIAL INEQUALITIES FOR MARTINGALES 128
9.3.1 EXPONENTIAL INEQUALITIES VIA DECOUPLING 128
9.3.2 CONDITIONALLY SYMMETRIC RANDOM VARIABLES 132
9.3.3 EXPONENTIAL SUPERMARTINGALES AND ASSOCIATED INEQUALITIES.. 134
9.4 SUPPLEMENTARY RESULTS AND PROBLEMS 135
10 A GENERAL FRAMEWORK FOR SELF-NORMALIZATION
137
10.1 AN EXPONENTIAL FAMILY OF SUPERMARTINGALES ASSOCIATED
WITH SELF-NORMALIZATION 137
10.1.1 THE I.I.D. CASE AND ANOTHER DERIVATION OF
(3.8)
137
10.1.2 A REPRESENTATION OF SELF-NORMALIZED PROCESSES
AND ASSOCIATED EXPONENTIAL SUPERMARTINGALES 138
10.2 CANONICAL ASSUMPTIONS AND RELATED STOCHASTIC MODELS 139
10.3 CONTINUOUS-TIME MARTINGALE THEORY 140
10.3.1 DOOB-MEYER DECOMPOSITION AND LOCALLY
SQUARE-INTEGRABLE MARTINGALES 141
10.3.2 INEQUALITIES AND STOCHASTIC INTEGRALS 143
10.4 SUPPLEMENTARY RESULTS AND PROBLEMS 146
11 PSEUDO-MAXIMIZATION VIA METHOD OF MIXTURES
149
11.1 PSEUDO-MAXIMIZATION AND LAPLACE S METHOD 149
11.2 A CLASS OF MIXING DENSITIES 150
11.3 APPLICATION OF METHOD OF MIXTURES TO BOUNDARY CROSSING
PROBABILITIES 152
11.3.1 THE ROBBINS-SIEGMUND BOUNDARIES FOR BROWNIAN MOTION .
. 152
11.3.2 EXTENSIONS TO GENERAL SELF-NORMALIZED PROCESSES 154
11.4 SUPPLEMENTARY RESULTS AND PROBLEMS 157
12 MOMENT AND EXPONENTIAL INEQUALITIES FOR SELF-NORMALIZED
PROCESSES
161
12.1 INEQUALITIES OF CABALLERO, FERNANDEZ AND NUALART, GRAVERSEN
AND PESKIR, AND KIKUCHI 161
12.2 MOMENT BOUNDS VIA THE METHOD OF MIXTURES 164
12.2.1 GAUSSIAN MIXING DENSITIES 165
12.2.2 THE MIXING DENSITY FUNCTIONS IN SECT. 11.2 167
12.3 APPLICATIONS AND EXAMPLES 174
12.3.1 PROOF OF LEMMA 8.11 174
12.3.2 GENERALIZATIONS OF THEOREMS 12.1, 12.2 AND 12.3 175
X
J
J CONTENTS
12.3.3 MOMENT INEQUALITIES UNDER CANONICAL ASSUMPTION
FOR A RESTRICTED RANGE 176
12.4 SUPPLEMENTARY RESULTS AND PROBLEMS 177
13 LAWS OF THE ITERATED LOGARITHM FOR SELF-NORMALIZED
PROCESSES 179
13.1 STOUT S LIL FOR SELF-NORMALIZED MARTINGALES 179
13.2 A UNIVERSAL UPPER LIL 182
13.3 COMPACT LIL FOR SELF-NORMALIZED MARTINGALES 186
13.4 SUPPLEMENTARY RESULTS AND PROBLEMS 190
14 MULTIVARIATE SELF-NORMALIZED PROCESSES WITH MATRIX NORMALIZATION
.. 193
14.1 MULTIVARIATE EXTENSION OF CANONICAL ASSUMPTIONS 193
14.1.1 MATRIX SEQUENCE ROOTS FOR SELF-NORMALIZATION 193
14.1.2 CANONICAL ASSUMPTIONS FOR MATRIX-NORMALIZED PROCESSES .
. 194
14.2 MOMENT AND EXPONENTIAL INEQUALITIES VIA PSEUDO-MAXIMIZATION ..
. 196
14.3 LIL AND BOUNDARY CROSSING PROBABILITIES FOR MULTIVARIATE
SELF-NORMALIZED PROCESSES 201
14.4 SUPPLEMENTARY RESULTS AND PROBLEMS 202
PART III STATISTICAL APPLICATIONS
15 THE/-STATISTIC AND STUDENTIZED STATISTICS
207
15.1 DISTRIBUTION THEORY OF STUDENT S /-STATISTICS 207
15.1.1 CASE OF INFINITE SECOND MOMENT 208
15.1.2 SADDLEPOINT APPROXIMATIONS 210
15.1.3 THE /-TEST AND A SEQUENTIAL EXTENSION 212
15.2 MULTIVARIATE EXTENSION AND HOTELLING S 7
2
-STATISTIC 213
15.2.1 SAMPLE COVARIANCE MATRIX AND WISHART DISTRIBUTION 213
15.2.2 THE MULTIVARIATE /-DISTRIBUTION AND HOTELLING S
^-STATISTI
C 213
15.2.3 ASYMPTOTIC THEORY IN THE CASE OF NON-NORMAL
Y
T
215
15.3 GENERAL STUDENTIZED STATISTICS 216
15.3.1 MARTINGALE CENTRAL LIMIT THEOREMS AND ASYMPTOTIC
NORMALITY 216
15.3.2 NON-NORMAL LIMITING DISTRIBUTIONS IN UNIT-ROOT
NONSTATIONARY AUTOREGRESSIVE MODELS 217
15.3.3 STUDENTIZED STATISTICS IN STOCHASTIC REGRESSION MODELS 218
15.4 SUPPLEMENTARY RESULTS AND PROBLEMS 221
16 SELF-NORMALIZATION FOR APPROXIMATE PIVOTS IN BOOTSTRAPPING
223
16.1 APPROXIMATE PIVOTS AND BOOTSTRAP-/ CONFIDENCE INTERVALS 223
16.2 EDGEWORTH EXPANSIONS AND SECOND-ORDER ACCURACY 224
16.2.1 EDGEWORTH EXPANSIONS FOR SMOOTH FUNCTIONS
OF SAMPLE MEANS 224
16.2.2 EDGEWORTH AND CORNISH-FISHER EXPANSIONS: APPLICATIONS
TO BOOTSTRAP-/ AND PERCENTILE INTERVALS 225
CONTENTS XIII
16.3 ASYMPTOTIC {/-STATISTICS AND THEIR BOOTSTRAP DISTRIBUTIONS 228
16.4 APPLICATION OF CRAMER-TYPE MODERATE DEVIATIONS 232
16.5 SUPPLEMENTARY RESULTS AND PROBLEMS 233
17 PSEUDO-MAXIMIZATION IN LIKELIHOOD AND BAYESIAN INFERENCE
235
17.1 GENERALIZED LIKELIHOOD RATIO STATISTICS 235
17.1.1 THE WILKS AND WALD STATISTICS 236
17.1.2 SCORE STATISTICS AND THEIR MARTINGALE PROPERTIES 238
17.2 PENALIZED LIKELIHOOD AND BAYESIAN INFERENCE 238
17.2.1 SCHWARZ S BAYESIAN SELECTION CRITERION 239
17.2.2 PSEUDO-MAXIMIZATION AND FREQUENTIST PROPERTIES
OF BAYES PROCEDURES 240
17.3 SUPPLEMENTARY RESULTS AND PROBLEMS 241
18 SEQUENTIAL ANALYSIS AND BOUNDARY CROSSING PROBABILITIES
FOR SELF-NORMALIZED STATISTICS
243
18.1 INFORMATION BOUNDS AND ASYMPTOTIC OPTIMALITY OF SEQUENTIAL
GLR TESTS 244
18.1.1 LIKELIHOOD RATIO IDENTITIES, THE WALD-HOEFFDING LOWER
BOUNDS AND THEIR ASYMPTOTIC GENERALIZATIONS 244
18.1.2 ASYMPTOTIC OPTIMALITY OF 2-SPRTS AND SEQUENTIAL GLR
TESTS 247
18.2 ASYMPTOTIC APPROXIMATIONS VIA METHOD OF MIXTURES
AND GEOMETRIC INTEGRATION 251
18.2.1 BOUNDARY CROSSING PROBABILITIES FOR GLR STATISTICS
VIA METHOD OF MIXTURES 251
18.2.2 A MORE GENERAL APPROACH USING SADDLEPOINT
APPROXIMATIONS AND GEOMETRIC INTEGRATION 252
18.2.3 APPLICATIONS AND EXAMPLES 257
18.3 EFFICIENT MONTE CARLO EVALUATION OF BOUNDARY
CROSSING PROBABILITIES 260
18.4 SUPPLEMENTARY RESULTS AND PROBLEMS 262
REFERENCES
267
INDEX
273
|
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| id | DE-604.BV035683117 |
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| indexdate | 2024-07-09T21:43:20Z |
| institution | BVB |
| isbn | 9783540856351 |
| language | English |
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| record_format | marc |
| series2 | Probability and its applications |
| spelling | De la Peña, Víctor H. Verfasser aut Self-normalized processes limit theory and statistical applications Victor H. de la Peña ; Tze Leung Lai ; Qi-Man Shao Self normalized processes Berlin [u.a.] Springer 2009 XIII, 275 S. txt rdacontent n rdamedia nc rdacarrier Probability and its applications Limit theorems (Probability theory) Mathematical statistics t-test (Statistics) t-Verteilung (DE-588)4300601-2 gnd rswk-swf Grenzwertsatz (DE-588)4158163-5 gnd rswk-swf U-Statistik (DE-588)4754777-7 gnd rswk-swf t-Verteilung (DE-588)4300601-2 s Grenzwertsatz (DE-588)4158163-5 s U-Statistik (DE-588)4754777-7 s DE-604 Lai, Tze Leung 1945- Verfasser (DE-588)135683971 aut Shao, Qi-Man Verfasser (DE-588)137328605 aut Erscheint auch als Online-Ausgabe 978-3-540-85636-8 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017737352&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | De la Peña, Víctor H. Lai, Tze Leung 1945- Shao, Qi-Man Self-normalized processes limit theory and statistical applications Limit theorems (Probability theory) Mathematical statistics t-test (Statistics) t-Verteilung (DE-588)4300601-2 gnd Grenzwertsatz (DE-588)4158163-5 gnd U-Statistik (DE-588)4754777-7 gnd |
| subject_GND | (DE-588)4300601-2 (DE-588)4158163-5 (DE-588)4754777-7 |
| title | Self-normalized processes limit theory and statistical applications |
| title_alt | Self normalized processes |
| title_auth | Self-normalized processes limit theory and statistical applications |
| title_exact_search | Self-normalized processes limit theory and statistical applications |
| title_full | Self-normalized processes limit theory and statistical applications Victor H. de la Peña ; Tze Leung Lai ; Qi-Man Shao |
| title_fullStr | Self-normalized processes limit theory and statistical applications Victor H. de la Peña ; Tze Leung Lai ; Qi-Man Shao |
| title_full_unstemmed | Self-normalized processes limit theory and statistical applications Victor H. de la Peña ; Tze Leung Lai ; Qi-Man Shao |
| title_short | Self-normalized processes |
| title_sort | self normalized processes limit theory and statistical applications |
| title_sub | limit theory and statistical applications |
| topic | Limit theorems (Probability theory) Mathematical statistics t-test (Statistics) t-Verteilung (DE-588)4300601-2 gnd Grenzwertsatz (DE-588)4158163-5 gnd U-Statistik (DE-588)4754777-7 gnd |
| topic_facet | Limit theorems (Probability theory) Mathematical statistics t-test (Statistics) t-Verteilung Grenzwertsatz U-Statistik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017737352&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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