Nonparametric goodness-of-fit testing under Gaussian models:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2003
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| Schriftenreihe: | Lecture notes in statistics
169 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 452 S. graph. Darst. |
| ISBN: | 0387955313 9780387955315 |
Internformat
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| 100 | 1 | |a Ingster, Yuri I. |e Verfasser |0 (DE-588)1051228050 |4 aut | |
| 245 | 1 | 0 | |a Nonparametric goodness-of-fit testing under Gaussian models |c Yu I. Ingster, I. A. Suslina |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2003 | |
| 300 | |a XIV, 452 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Lecture notes in statistics |v 169 | |
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| 650 | 7 | |a Statistische toetsen |2 gtt | |
| 650 | 4 | |a Goodness-of-fit tests | |
| 650 | 4 | |a Nonparametric statistics | |
| 650 | 0 | 7 | |a Güte der Anpassung |0 (DE-588)4202143-1 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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| adam_text | YU. I. INGSTER LA. SUSLINA NONPARAMETRIC GOODNESS-OF-FIT TESTING UNDER
GAUSSIAN MODELS SPRINGER CONTENTS PREFACE V 1 INTRODUCTION 1 1.1 TESTS 4
1.2 ONE-DIMENSIONAL PARAMETER 5 1.3 MULTIDIMENSIONAL PARAMETER 9 1.4
INFINITE-DIMENSIONAL PARAMETER . . . . * 15 1.4.1 CLASSICAL
GOODNESS-OF-FIT TESTS 15 1.4.2 NONDISTINGUISHABILITY IN THE PROBLEM 22
1.4.3 TWO-SIDED CONSTRAINT PROBLEM 23 1.4.4 COMPARING WITH THE MINIMAX
ESTIMATION PROBLEM 25 1.5 PROBLEMS OF THE STUDY AND MAIN RESULTS 28 1.6
METHODS OF THE STUDY 32 1.7 STRUCTURE OF THE BOOK 35 2 AN OVERVIEW 38
2.1 MODELS . 38 2.1.1 GAUSSIAN MODELS IN SEQUENCE SPACE 38 2.1.2
FUNCTIONAL GAUSSIAN MODEL 39 2.2 HYPOTHESIS TESTING PROBLEM 40 2.2.1
HYPOTHESIS, ALTERNATIVE, TESTS 40 2.2.2 HOW TO COMPARE THE TESTS? 41
2.2.3 SIMPLE HYPOTHESIS AND AN ALTERNATIVE 42 2.2.4 LI-DISTANCE ON THE
SPACE OF MEASURES 44 CONTENTS 2.3 BAYESIAN APPROACH IN HYPOTHESIS
TESTING 47 2.3.1 GENERAL SETTING 47 2.3.2 BAYESIAN APPROACH IN THE
GAUSSIAN MODEL . . . . 48 2.4 MINIMAX APPROACH IN HYPOTHESIS TESTING 53
2.4.1 GENERAL SETTING . . . , 53 2.4.2 CONNECTION WITH THE BAYESIAN
APPROACH 55 2.4.3 CONVEX HYPOTHESIS AND AN ALTERNATIVE UNDER THE
GAUSSIAN MODEL 58 2.4.4 CONVEX SYMMETRIC ADMISSIBLE SET 61 2.5
ASYMPTOTICS IN HYPOTHESIS TESTING 62 2.5.1 ASYMPTOTIC APPROACH 62 2.5.2
ASYMPTOTIC MINIMAXITY 63 2.5.3 ASYMPTOTIC MINIMAXITY IN THE PARAMETRIC
CASE . 65 2.6 MINIMAX DISTINGUISHABILITY IN GOODNESS-OF-FIT PROBLEMS 67
2.6.1 THE PARAMETRIC CASE 68 2.6.2 * CLASSICAL DISTINGUISHABILITY IN THE
NONPARAMETRIC CASE 68 2.6.3 NONDISTINGUISHABILITY IN THE NONPARAMETRIC
CASE 69 2.7 NORMS AND WAVELET TRANSFORM 71 2.7.1 NORMS IN THE SEQUENCE
SPACE 72 2.7.2 NORMS IN THE FUNCTIONAL SPACE 73 2.7.3 THE WAVELET
TRANSFORM 76 2.8 SHORT OVERVIEW OF MINIMAX ESTIMATION 77 2.9 CONSTRAINTS
OF INTEREST 79 2.10 RATES IN ESTIMATION AND IN HYPOTHESIS TESTING 79
MINIMAX DISTINGUISHABILITY 81 3.1 MINIMAX PROPERTIES OF TEST FAMILIES-
81 3.1.1 CONSISTENCY CONDITIONS AND GAUSSIAN BOUNDS . . 81 3.1.2 TESTS
OF X 2 -TYPE 83 3.1.3 TESTS OF X P -TYPE 85 3.1.4 SUPREME TESTS 89 3.2
ASYMPTOTIC MINIMAXITY FOR SQUARE NORMS . . . . . . . . . 92 ,3.2.1 LOWER
BOUNDS AND INNER RADII 93 3.2.2 UPPER BOUNDS AND KOLMOGOROV DIAMETERS .
. . . 95 3.3 BAYESIAN APPROACH UNDER A GAUSSIAN MODEL 97 3.3.1 L - AND L
2 -DISTANCES FOR MIXTURES 97 3.3.2 PRODUCT PRIORS . . 98 3.3.3 HILBERT
STRUCTURE 99 3.3.4 TWO-POINT FACTORS. 102 3.3.5 SYMMETRIC THREE-POINT
FACTORS 103 3.3.6 MINIMAX DISTINGUISHABILITY FOR /P-BALLS, P OO . 104
3.3.7 SEMIORTHOGONAL PRIORS 106 3.3.8 MINIMAX DISTINGUISHABILITY FOR
Z-BALLS, P = OO . 108 3.4 TRIVIALITY AND CLASSICAL ASYMPTOTICS 109
CONTENTS XI 3.4.1 MINIMAX TRIVIALITY 109 3.4.2 TRIVIALITY FOR Z P -BALLS
109 3.4.3 TRIVIALITY AND CLASSICAL ASYMPTOTICS FOR POWER NORMS 110 3.4.4
TRIVIALITY AND CLASSICAL ASYMPTOTICS FOR BESOV NORMS 115 3.4.5
TRIVIALITY AND CLASSICAL ASYMPTOTICS FOR A POSITIVE ALTERNATIVE 116
3.4.6 NONEMPTY AND TRIVIALITY FOR POWER NORMS: TWO- SIDED CONSTRAINTS
117 3.4.7 TRIVIALITY FOR BESOV NORMS: TWO-SIDED CONSTRAINTS 123 3.4.8
TRIVIALITY UNDER THE FUNCTIONAL GAUSSIAN MODEL . 127 3.5
DISTINGUISHABILITY CONDITIONS FOR SMOOTH SIGNALS . . . . 128 3.5.1 RATES
AND TESTS FOR SMOOTH SIGNALS 128 3.5.2 PROOF OF THE LOWER BOUNDS 130
3.5.3 PROOF OF THE UPPER BOUNDS 134 3.5.4 RATES FOR POSITIVE SMOOTH
SIGNALS 135 4 SHARP ASYMPTOTICS. I 136 4.1 TESTS BASED ON LINEAR
STATISTICS AND CONVEX ALTERNATIVES 137 4.2 TWO-SIDED CONSTRAINTS FOR THE
POSITIVE ALTERNATIVES, P L,Q P 141 4.2.1 SYSTEM OF EQUATIONS FOR AN
EXTREME PROBLEM . . 142 4.2.2 SOLUTION OF AN EXTREME PROBLEM: CASE P Q
OO 144 4.2.3 SOLUTION OF AN EXTREME PROBLEM: CASE P = Q . . . 146
4.2.4 SOLUTION OF AN EXTREME PROBLEM: CASE P Q = OO 147 4.2.5
CONSTRAINTS OF EXPONENTIAL TYPE WITH P 1, Q P 150 4.3 SHARP
ASYMPTOTICS OF GAUSSIAN TYPE: PRODUCT PRIORS . . 154 4.3.1 ASYMPTOTIC
NORMALITY FOR PRODUCT PRIORS . . . . 155 4.3.2 SYMMETRIC TWO-POINT
FACTORS AND TESTS OF % 2 -TYPE 159 4.3.3 THE POWER NORMS: P 2, Q P
162 4.3.4 BESOV NORMS: P 2, Q P V. . 168 4.3.5 SMOOTH SIGNALS: CASE P
2, Q P . 171 4.4 SHARP ASYMPTOTICS: ASYMPTOTIC DEGENERACY 173 4.4.1
ASYMPTOTIC DEGENERACY IN BAYESIAN PROBLEMS . . 173 4.4.2 DEGENERATE
ASYMPTOTICS FOR POWER NORMS . . . . 176 4.4.3 DEGENERATE ASYMPTOTICS FOR
BESOV NORMS . . . . 180 4.4.4 DEGENERATE ASYMPTOTICS FOR SMOOTH SIGNALS
... 184 5 SHARP ASYMPTOTICS. II 185 5.1 TESTS BASED ON LOG-LIKELIHOOD
STATISTICS AND THRESHOLDING 185 5.1.1 TEST PROCEDURES 185 5.1.2
REDUCTION TO THE NONTRUNCATED CASE 191 5.2 EXTREME PROBLEM IN THE SPACE
OF SEQUENCES OF MEASURES 194 5.2.1 STATEMENT OF THE PROBLEM 194 XII
CONTENTS 5.2.2 MOMENT CONSTRAINTS 196 5.2.3 CONVEX PROPERTIES OF THE
EXTREME PROBLEM . . . 200, 5.3 SEPARATION OF THE PROBLEM 203 5.3.1 CASE
P = Q 204 5.3.2 CASEP^G 204 5.4 SOLUTION OF ONE-DIMENSIONAL PROBLEMS 208
5.4.1 LEMMA ON CONVEXITY 209 5.4.2 CASEP = G 212 5.4.3 CASEP^G 214 5.4.4
REDUCTION OF THE EXTREME PROBLEMS (5.76), (5.85) 218 5.5 SHARP
ASYMPTOTICS FOR / -BALLS 224 5.5.1 ONE-SIDED CONSTRAINT 224 5.5.2
TWO-SIDED CONSTRAINTS 226 6 GAUSSIAN ASYMPTOTICS FOR POWER AND BESOV
NORMS 231 6.1 EXTREME PROBLEMS 232 6.2 PRINCIPAL TYPES OF GAUSSIAN
ASYMPTOTICS 234 6.2.1 PRINCIPAL TYPES OF GAUSSIAN ASYMPTOTICS FOR POWER
NORMS 235 6.2.2 PRINCIPAL TYPES OF GAUSSIAN ASYMPTOTICS FOR BESOV ,
NORMS 237 6.3 FRONTIER LOG-TYPES OF GAUSSIAN ASYMPTOTICS 240 6.3.1
FRONTIER TYPES OF ASYMPTOTICS FOR POWER NORMS . 240 6.3.2 FRONTIER TYPES
OF ASYMPTOTICS FOR BESOV NORMS . 241 6.4 GRAPHICAL PRESENTATION 243 -
6.5 REMARKS ON THE PROOFS OF THEOREMS 6.1-6.4 246 6.6 PROOF OF THEOREMS
6.1 AND 6.3 FOR P 2, Q P, AND P = Q 249 6.6.1 CASE P 2, Q P . *:
249 -_ 6.6.2 CASEP = 9 2 250 6.7 EXTREME PROBLEM FOR POWER NORMS: P ^ Q
253 6.7.1 EQUATIONS SYSTEM FOR Q OO 254 6.7.2 EQUATIONS SYSTEM FOR Q =
OO .-.... 255 6.8 PROPERTIES OF THE EXTREME SEQUENCES FOR POWER NORMS .
256 6.8.1 CONTINUITY PROPERTIES 257 6.8.2 CORRELATION PROPERTIES 258 6.9
EXTREME PROBLEM FOR BESOV NORMS 258 7 ADAPTATION FOR POWER AND BESOV
NORMS 262 7.1 ADAPTIVE SETTING 263 7.1.1 IDEA OF THE STUDY 263 7.1.2
STATEMENT OF THE PROBLEM 265 7.1.3 MAIN RESULTS 266 7.2 LOWER BOUNDS 268
7.2.1 ASYMPTOTICALLY ORTHOGONAL PRIORS 269 7.2.2 ASYMPTOTIC SHARP LOWER
BOUNDS 270 CONTENTS XIII 7.2.3 LOWER BOUNDS FOR POWER NORMS 276 7.2.4
NONDISTINGUISHABILITY CONDITIONS FOR BESOV NORMS 278 7.3 UPPER BOUNDS
FOR POWER NORMS 280 7.3.1 METHODS OF CONSTRUCTIONS . 280 7.3.2 THE
CONSTRUCTION OF COLLECTIONS OF TESTS 282 7.4 UPPER BOUNDS FOR BESOV
NORMS 285 7.4.1 TEST PROCEDURE 285 7.4.2 TYPE I ERRORS 287 7.4.3 TYPE II
ERRORS FOR PARTICULAR TESTS 288 HIGH-DIMENSIONAL SIGNAL DETECTION 291
8.1 THE BAYESIAN SIGNAL DETECTION PROBLEM 292 8.1.1 LIMIT THEOREMS AND
INFINITELY DIVISIBLE DISTRIBU- TIONS 293 8.1.2 SOME REMARKS ON THE STUDY
OF LOG-LIKELIHOOD RATIOS 294 8.1.3 LIMIT DISTRIBUTIONS FOR R 2 297
8.1.4 LIMIT DISTRIBUTIONS FOR T G (1,2) 299 8.1.5 LIMIT DISTRIBUTIONS
FOR R 1 304 8.1.6 GRAPHIC REPRESENTATION 306 8.2 MULTICHANNEL SIGNAL
DETECTION PROBLEMS 307 8.2.1 REDUCTION TO THE BAYESIAN PROBLEM 308 8.2.2
PROOF OF THEOREM 8.1 309 8.3 MINIMAX SIGNAL DETECTION FOR Z N -BALLS 314
8.3.1 MAIN RESULTS 314 8.3.2 EXAMPLES 316 8.3.3 PROOF OF LOWER BOUNDS
318 8.4 PROOF OF UPPER BOUNDS - 319 8.4.1 UPPER BOUNDS FOR R 2 321
8.4.2 UPPER BOUNDS FOR T G (1,2) 325 8.4.3 STUDY OF THE TESTS (8.41) 334
8.5 TESTING A HYPOTHESIS WHICH IS CLOSE TO A SIMPLE HYPOTHESIS 337 8.5.1
STATEMENT AND MAIN RESULTS 337 8.5.2 PROOF OF LOWER BOUNDS 338 8.5.3
UPPER BOUNDS 344 8.5.4 EXTENSION TO THE FUNCTIONAL GAUSSIAN MODEL . . .
350 L APPENDIX 354 A.I PROOF OF PROPOSITION 2.16 354 A.2 PROOF OF
PROPOSITION 5.3 356 A.2.1 PROPERTIES OF STATISTICS UNDER ALTERNATIVES
356 A.2.2 EVALUATIONS OF TYPE II ERRORS 359 A.3 STUDY OF THE EXTREME
PROBLEM FOR POWER NORMS 364 A.3.1 SOLUTION OF THE SYSTEM (6.86), (6.87)
364 A.3.2 SOLUTIONS OF THE EXTREME PROBLEM FOR K H^ . . 369 XIV
CONTENTS A.3.3 PROOFS OF THEOREMS 6.1, 6.3 FOR K G H^ 372 A.3.4 SOLUTION
OF THE EXTREME PROBLEM (6.88) 376 A.3.5 SOLUTION OF THE EXTREME PROBLEM
FOR K G SJ . . . 379 A.3.6 PROOFS OF THEOREMS 6.1, 6.3 FOR K G SJ 385
A.3.7 EXTREME PROBLEM FOR Q = OO .389 A.3.8 PROOFS OF PROPOSITIONS 6.1,
6.2 391 A.4 STUDY OF THE EXTREME PROBLEM FOR BESOV NORMS 396 A.4.1
SOLUTION OF THE SYSTEM (6.110), (6.111) 396 A.4.2 SOLUTION OF THE
EXTREME PROBLEM (6.112) 402 A.4.3 SOLUTION OF THE EXTREME PROBLEM FOR R
G T^ . . 407 A.4.4 SOLUTION OF THE EXTREME PROBLEM FOR R G T% T . . 409
A.4.5 UPPER BOUNDS 419 A.4.6 LOWER BOUNDS 425 A.4.7 PROOF OF PROPOSITION
6.3 432 A.5 PROOF OF LEMMA 7.4 433 A.5.1 CASE K E GL 436 A.5.2 CASE K
G HG 2 438 A.6 PROOFS OF LEMMAS 8.2, 8.3, 8.4, 8.6 439 REFERENCES 444
INDEX 450 SUBJECT INDEX 450 PARAMETER AND FUNCTION INDEX 453
|
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| ctrlnum | (OCoLC)49936099 (DE-599)BVBBV014527109 |
| dewey-full | 519.5/4 519.5/421 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5/4 519.5/4 21 |
| dewey-search | 519.5/4 519.5/4 21 |
| dewey-sort | 3519.5 14 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV014527109 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:03:15Z |
| institution | BVB |
| isbn | 0387955313 9780387955315 |
| language | English |
| lccn | 2002070731 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009887683 |
| oclc_num | 49936099 |
| open_access_boolean | |
| owner | DE-739 DE-19 DE-BY-UBM DE-824 DE-91G DE-BY-TUM DE-521 DE-83 DE-11 |
| owner_facet | DE-739 DE-19 DE-BY-UBM DE-824 DE-91G DE-BY-TUM DE-521 DE-83 DE-11 |
| physical | XIV, 452 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in statistics |
| series2 | Lecture notes in statistics |
| spelling | Ingster, Yuri I. Verfasser (DE-588)1051228050 aut Nonparametric goodness-of-fit testing under Gaussian models Yu I. Ingster, I. A. Suslina New York [u.a.] Springer 2003 XIV, 452 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in statistics 169 Non-parametrische statistiek gtt Normale verdeling gtt Statistische modellen gtt Statistische toetsen gtt Goodness-of-fit tests Nonparametric statistics Güte der Anpassung (DE-588)4202143-1 gnd rswk-swf Nichtparametrischer Test (DE-588)4075372-4 gnd rswk-swf Güte der Anpassung (DE-588)4202143-1 s Nichtparametrischer Test (DE-588)4075372-4 s DE-604 Suslina, Irina A. Sonstige oth Erscheint auch als Online-Ausgabe 978-0-387-21580-8 Lecture notes in statistics 169 (DE-604)BV002447846 169 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009887683&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ingster, Yuri I. Nonparametric goodness-of-fit testing under Gaussian models Lecture notes in statistics Non-parametrische statistiek gtt Normale verdeling gtt Statistische modellen gtt Statistische toetsen gtt Goodness-of-fit tests Nonparametric statistics Güte der Anpassung (DE-588)4202143-1 gnd Nichtparametrischer Test (DE-588)4075372-4 gnd |
| subject_GND | (DE-588)4202143-1 (DE-588)4075372-4 |
| title | Nonparametric goodness-of-fit testing under Gaussian models |
| title_auth | Nonparametric goodness-of-fit testing under Gaussian models |
| title_exact_search | Nonparametric goodness-of-fit testing under Gaussian models |
| title_full | Nonparametric goodness-of-fit testing under Gaussian models Yu I. Ingster, I. A. Suslina |
| title_fullStr | Nonparametric goodness-of-fit testing under Gaussian models Yu I. Ingster, I. A. Suslina |
| title_full_unstemmed | Nonparametric goodness-of-fit testing under Gaussian models Yu I. Ingster, I. A. Suslina |
| title_short | Nonparametric goodness-of-fit testing under Gaussian models |
| title_sort | nonparametric goodness of fit testing under gaussian models |
| topic | Non-parametrische statistiek gtt Normale verdeling gtt Statistische modellen gtt Statistische toetsen gtt Goodness-of-fit tests Nonparametric statistics Güte der Anpassung (DE-588)4202143-1 gnd Nichtparametrischer Test (DE-588)4075372-4 gnd |
| topic_facet | Non-parametrische statistiek Normale verdeling Statistische modellen Statistische toetsen Goodness-of-fit tests Nonparametric statistics Güte der Anpassung Nichtparametrischer Test |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009887683&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002447846 |
| work_keys_str_mv | AT ingsteryurii nonparametricgoodnessoffittestingundergaussianmodels AT suslinairinaa nonparametricgoodnessoffittestingundergaussianmodels |