Testing statistical hypotheses:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer
2005
|
| Ausgabe: | 3. ed. |
| Schriftenreihe: | Springer texts in statistics
|
| Schlagworte: | |
| Online-Zugang: | BTU01 TUM01 UBA01 UBR01 UBT01 Volltext Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | 1 Online-Ressource (XIV, 784 S.) graph. Darst. |
| ISBN: | 9780387276052 |
| DOI: | 10.1007/0-387-27605-X |
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Datensatz im Suchindex
| _version_ | 1804136428847759360 |
|---|---|
| adam_text | Contents
Preface vii
I Small Sample Theory 1
1 The General Decision Problem 3
1.1 Statistical Inference and Statistical Decisions 3
1.2 Specification of a Decision Problem 4
1.3 Randomization; Choice of Experiment 8
1.4 Optimum Procedures 9
1.5 Invariance and Unbiasedness 11
1.6 Bayes and Minimax Procedures 14
1.7 Maximum Likelihood 16
1.8 Complete Classes 17
1.9 Sufficient Statistics 18
1.10 Problems 21
1.11 Notes 27
2 The Probability Background 28
2.1 Probability and Measure 28
2.2 Integration 31
2.3 Statistics and Subfields 34
2.4 Conditional Expectation and Probability 36
2.5 Conditional Probability Distributions 41
2.6 Characterization of Sufficiency 44
2.7 Exponential Families 46
x Contents
2.8 Problems 50
2.9 Notes 55
3 Uniformly Most Powerful Tests 56
3.1 Stating The Problem 56
3.2 The Neyman Pearson Fundamental Lemma 59
3.3 p values 63
3.4 Distributions with Monotone Likelihood Ratio 65
3.5 Confidence Bounds 72
3.6 A Generalization of the Fundamental Lemma 77
3.7 Two Sided Hypotheses 81
3.8 Least Favorable Distributions 83
3.9 Applications to Normal Distributions 86
3.9.1 Univariate Normal Models 86
3.9.2 Multivariate Normal Models 89
3.10 Problems 92
3.11 Notes 107
4 Unbiasedness: Theory and First Applications 110
4.1 Unbiasedness For Hypothesis Testing 110
4.2 One Parameter Exponential Families Ill
4.3 Similarity and Completeness 115
4.4 UMP Unbiased Tests for Multiparameter Exponential Families 119
4.5 Comparing Two Poisson or Binomial Populations 124
4.6 Testing for Independence in a 2 x 2 Table 127
4.7 Alternative Models for 2 x 2 Tables 130
4.8 Some Three Factor Contingency Tables 132
4.9 The Sign Test 135
4.10 Problems 139
4.11 Notes 149
5 Unbiasedness: Applications to Normal Distributions 150
5.1 Statistics Independent of a Sufficient Statistic 150
5.2 Testing the Parameters of a Normal Distribution 153
5.3 Comparing the Means and Variances of Two Normal Distribu¬
tions 157
5.4 Confidence Intervals and Families of Tests 161
5.5 Unbiased Confidence Sets 164
5.6 Regression 168
5.7 Bayesian Confidence Sets 171
5.8 Permutation Tests 176
5.9 Most Powerful Permutation Tests 177
5.10 Randomization As A Basis For Inference 181
5.11 Permutation Tests and Randomization 184
5.12 Randomization Model and Confidence Intervals 187
5.13 Testing for Independence in a Bivariate Normal Distribution . 190
5.14 Problems 192
5.15 Notes 210
Contents xi
6 Invar iance 212
6.1 Symmetry and Invariance 212
6.2 Maximal Invariants 214
6.3 Most Powerful Invariant Tests 218
6.4 Sample Inspection by Variables 223
6.5 Almost Invariance 225
6.6 Unbiasedness and Invariance 229
6.7 Admissibility 232
6.8 Rank Tests 239
6.9 The Two Sample Problem 242
6.10 The Hypothesis of Symmetry 246
6.11 Equivariant Confidence Sets 248
6.12 Average Smallest Equivariant Confidence Sets 251
6.13 Confidence Bands for a Distribution Function 255
6.14 Problems 257
6.15 Notes 276
7 Linear Hypotheses 277
7.1 A Canonical Form 277
7.2 Linear Hypotheses and Least Squares 281
7.3 Tests of Homogeneity 285
7.4 Two Way Layout: One Observation per Cell 287
7.5 Two Way Layout: m Observations Per Cell 290
7.6 Regression 293
7.7 Random Effects Model: One way Classification 297
7.8 Nested Classifications 300
7.9 Multivariate Extensions 304
7.10 Problems 306
7.11 Notes 317
8 The Minimax Principle 319
8.1 Tests with Guaranteed Power 319
8.2 Examples 322
8.3 Comparing Two Approximate Hypotheses 326
8.4 Maximin Tests and Invariance 329
8.5 The Hunt Stein Theorem 331
8.6 Most Stringent Tests 337
8.7 Problems 338
8.8 Notes 347
9 Multiple Testing and Simultaneous Inference 348
9.1 Introduction and the FWER 348
9.2 Maximin Procedures 354
9.3 The Hypothesis of Homogeneity 363
9.4 Scheffe s S Method: A Special Case 375
9.5 Scheffe s 5 Method for General Linear Models 380
9.6 Problems 385
9.7 Notes 391
xii Contents
10 Conditional Inference 392
10.1 Mixtures of Experiments 392
10.2 Ancillary Statistics 395
10.3 Optimal Conditional Tests 400
10.4 Relevant Subsets 404
10.5 Problems 409
10.6 Notes 414
11 Large Sample Theory 417
11 Basic Large Sample Theory 419
11.1 Introduction 419
11.2 Basic Convergence Concepts 424
11.2.1 Weak Convergence and Central Limit Theorems . . . 424
11.2.2 Convergence in Probability and Applications 431
11.2.3 Almost Sure Convergence 440
11.3 Robustness of Some Classical Tests 444
11.3.1 Effect of Distribution 444
11.3.2 Effect of Dependence 448
11.3.3 Robustness in Linear Models 451
11.4 Nonparametric Mean 459
11.4.1 Edgeworth Expansions 459
11.4.2 Thet test 462
11.4.3 A Result of Bahadur and Savage 466
11.4.4 Alternative Tests 468
11.5 Problems 469
11.6 Notes 480
12 Quadratic Mean Differentiable Families 482
12.1 Introduction 482
12.2 Quadratic Mean Differentiability (q.m.d.) 482
12.3 Contiguity 492
12.4 Likelihood Methods in Parametric Models 503
12.4.1 Efficient Likelihood Estimation 504
12.4.2 Wald Tests and Confidence Regions 508
12.4.3 Rao Score Tests 511
12.4.4 Likelihood Ratio Tests 513
12.5 Problems 517
12.6 Notes 525
13 Large Sample Optimality 527
13.1 Testing Sequences, Metrics, and Inequalities 527
13.2 Asymptotic Relative Efficiency 534
13.3 AUMP Tests in Univariate Models 540
13.4 Asymptotically Normal Experiments 549
13.5 Applications to Parametric Models 553
13.5.1 One sided Hypotheses 553
13.5.2 Equivalence Hypotheses 559
Contents xiii
13.5.3 Multi sided Hypotheses 564
13.6 Applications to Nonparametric Models 567
13.6.1 Nonparametric Mean 567
13.6.2 Nonparametric Testing of Punctionals 570
13.7 Problems 574
13.8 Notes 582
14 Testing Goodness of Fit 583
14.1 Introduction 583
14.2 The Kolmogorov Smirnov Test 584
14.2.1 Simple Null Hypothesis 584
14.2.2 Extensions of the Kolmogorov Smirnov Test 589
14.3 Pearson s Chi squared Statistic 590
14.3.1 Simple Null Hypothesis 590
14.3.2 Chi squared Test of Uniformity 594
14.3.3 Composite Null Hypothesis 597
14.4 Neyman s Smooth Tests 599
14.4.1 Fixed k Asymptotics 601
14.4.2 Neyman s Smooth Tests With Large k 603
14.5 Weighted Quadratic Test Statistics 607
14.6 Global Behavior of Power Functions 616
14.7 Problems 622
14.8 Notes 629
15 General Large Sample Methods 631
15.1 Introduction 631
15.2 Permutation and Randomization Tests 632
15.2.1 The Basic Construction 632
15.2.2 Asymptotic Results 636
15.3 Basic Large Sample Approximations 643
15.3.1 Pivotal Method 644
15.3.2 Asymptotic Pivotal Method 646
15.3.3 Asymptotic Approximation 647
15.4 Bootstrap Sampling Distributions 648
15.4.1 Introduction and Consistency 648
15.4.2 The Nonparametric Mean 653
15.4.3 Further Examples 655
15.4.4 Stepdown Multiple Testing 658
15.5 Higher Order Asymptotic Comparisons 661
15.6 Hypothesis Testing 668
15.7 Subsampling 673
15.7.1 The Basic Theorem in the I.I.D. Case 674
15.7.2 Comparison with the Bootstrap 677
15.7.3 Hypothesis Testing 680
15.8 Problems 682
15.9 Notes 690
A Auxiliary Results 692
A.I Equivalence Relations; Groups 692
xiv Contents
A.2 Convergence of Functions; Metric Spaces 693
A.3 Banach and Hilbert Spaces 696
A.4 Dominated Families of Distributions 698
A.5 The Weak Compactness Theorem 700
References 702
Author Index 757
Subject Index 767
|
| adam_txt |
Contents
Preface vii
I Small Sample Theory 1
1 The General Decision Problem 3
1.1 Statistical Inference and Statistical Decisions 3
1.2 Specification of a Decision Problem 4
1.3 Randomization; Choice of Experiment 8
1.4 Optimum Procedures 9
1.5 Invariance and Unbiasedness 11
1.6 Bayes and Minimax Procedures 14
1.7 Maximum Likelihood 16
1.8 Complete Classes 17
1.9 Sufficient Statistics 18
1.10 Problems 21
1.11 Notes 27
2 The Probability Background 28
2.1 Probability and Measure 28
2.2 Integration 31
2.3 Statistics and Subfields 34
2.4 Conditional Expectation and Probability 36
2.5 Conditional Probability Distributions 41
2.6 Characterization of Sufficiency 44
2.7 Exponential Families 46
x Contents
2.8 Problems 50
2.9 Notes 55
3 Uniformly Most Powerful Tests 56
3.1 Stating The Problem 56
3.2 The Neyman Pearson Fundamental Lemma 59
3.3 p values 63
3.4 Distributions with Monotone Likelihood Ratio 65
3.5 Confidence Bounds 72
3.6 A Generalization of the Fundamental Lemma 77
3.7 Two Sided Hypotheses 81
3.8 Least Favorable Distributions 83
3.9 Applications to Normal Distributions 86
3.9.1 Univariate Normal Models 86
3.9.2 Multivariate Normal Models 89
3.10 Problems 92
3.11 Notes 107
4 Unbiasedness: Theory and First Applications 110
4.1 Unbiasedness For Hypothesis Testing 110
4.2 One Parameter Exponential Families Ill
4.3 Similarity and Completeness 115
4.4 UMP Unbiased Tests for Multiparameter Exponential Families 119
4.5 Comparing Two Poisson or Binomial Populations 124
4.6 Testing for Independence in a 2 x 2 Table 127
4.7 Alternative Models for 2 x 2 Tables 130
4.8 Some Three Factor Contingency Tables 132
4.9 The Sign Test 135
4.10 Problems 139
4.11 Notes 149
5 Unbiasedness: Applications to Normal Distributions 150
5.1 Statistics Independent of a Sufficient Statistic 150
5.2 Testing the Parameters of a Normal Distribution 153
5.3 Comparing the Means and Variances of Two Normal Distribu¬
tions 157
5.4 Confidence Intervals and Families of Tests 161
5.5 Unbiased Confidence Sets 164
5.6 Regression 168
5.7 Bayesian Confidence Sets 171
5.8 Permutation Tests 176
5.9 Most Powerful Permutation Tests 177
5.10 Randomization As A Basis For Inference 181
5.11 Permutation Tests and Randomization 184
5.12 Randomization Model and Confidence Intervals 187
5.13 Testing for Independence in a Bivariate Normal Distribution . 190
5.14 Problems 192
5.15 Notes 210
Contents xi
6 Invar iance 212
6.1 Symmetry and Invariance 212
6.2 Maximal Invariants 214
6.3 Most Powerful Invariant Tests 218
6.4 Sample Inspection by Variables 223
6.5 Almost Invariance 225
6.6 Unbiasedness and Invariance 229
6.7 Admissibility 232
6.8 Rank Tests 239
6.9 The Two Sample Problem 242
6.10 The Hypothesis of Symmetry 246
6.11 Equivariant Confidence Sets 248
6.12 Average Smallest Equivariant Confidence Sets 251
6.13 Confidence Bands for a Distribution Function 255
6.14 Problems 257
6.15 Notes 276
7 Linear Hypotheses 277
7.1 A Canonical Form 277
7.2 Linear Hypotheses and Least Squares 281
7.3 Tests of Homogeneity 285
7.4 Two Way Layout: One Observation per Cell 287
7.5 Two Way Layout: m Observations Per Cell 290
7.6 Regression 293
7.7 Random Effects Model: One way Classification 297
7.8 Nested Classifications 300
7.9 Multivariate Extensions 304
7.10 Problems 306
7.11 Notes 317
8 The Minimax Principle 319
8.1 Tests with Guaranteed Power 319
8.2 Examples 322
8.3 Comparing Two Approximate Hypotheses 326
8.4 Maximin Tests and Invariance 329
8.5 The Hunt Stein Theorem 331
8.6 Most Stringent Tests 337
8.7 Problems 338
8.8 Notes 347
9 Multiple Testing and Simultaneous Inference 348
9.1 Introduction and the FWER 348
9.2 Maximin Procedures 354
9.3 The Hypothesis of Homogeneity 363
9.4 Scheffe's S Method: A Special Case 375
9.5 Scheffe's 5 Method for General Linear Models 380
9.6 Problems 385
9.7 Notes 391
xii Contents
10 Conditional Inference 392
10.1 Mixtures of Experiments 392
10.2 Ancillary Statistics 395
10.3 Optimal Conditional Tests 400
10.4 Relevant Subsets 404
10.5 Problems 409
10.6 Notes 414
11 Large Sample Theory 417
11 Basic Large Sample Theory 419
11.1 Introduction 419
11.2 Basic Convergence Concepts 424
11.2.1 Weak Convergence and Central Limit Theorems . . . 424
11.2.2 Convergence in Probability and Applications 431
11.2.3 Almost Sure Convergence 440
11.3 Robustness of Some Classical Tests 444
11.3.1 Effect of Distribution 444
11.3.2 Effect of Dependence 448
11.3.3 Robustness in Linear Models 451
11.4 Nonparametric Mean 459
11.4.1 Edgeworth Expansions 459
11.4.2 Thet test 462
11.4.3 A Result of Bahadur and Savage 466
11.4.4 Alternative Tests 468
11.5 Problems 469
11.6 Notes 480
12 Quadratic Mean Differentiable Families 482
12.1 Introduction 482
12.2 Quadratic Mean Differentiability (q.m.d.) 482
12.3 Contiguity 492
12.4 Likelihood Methods in Parametric Models 503
12.4.1 Efficient Likelihood Estimation 504
12.4.2 Wald Tests and Confidence Regions 508
12.4.3 Rao Score Tests 511
12.4.4 Likelihood Ratio Tests 513
12.5 Problems 517
12.6 Notes 525
13 Large Sample Optimality 527
13.1 Testing Sequences, Metrics, and Inequalities 527
13.2 Asymptotic Relative Efficiency 534
13.3 AUMP Tests in Univariate Models 540
13.4 Asymptotically Normal Experiments 549
13.5 Applications to Parametric Models 553
13.5.1 One sided Hypotheses 553
13.5.2 Equivalence Hypotheses 559
Contents xiii
13.5.3 Multi sided Hypotheses 564
13.6 Applications to Nonparametric Models 567
13.6.1 Nonparametric Mean 567
13.6.2 Nonparametric Testing of Punctionals 570
13.7 Problems 574
13.8 Notes 582
14 Testing Goodness of Fit 583
14.1 Introduction 583
14.2 The Kolmogorov Smirnov Test 584
14.2.1 Simple Null Hypothesis 584
14.2.2 Extensions of the Kolmogorov Smirnov Test 589
14.3 Pearson's Chi squared Statistic 590
14.3.1 Simple Null Hypothesis 590
14.3.2 Chi squared Test of Uniformity 594
14.3.3 Composite Null Hypothesis 597
14.4 Neyman's Smooth Tests 599
14.4.1 Fixed k Asymptotics 601
14.4.2 Neyman's Smooth Tests With Large k 603
14.5 Weighted Quadratic Test Statistics 607
14.6 Global Behavior of Power Functions 616
14.7 Problems 622
14.8 Notes 629
15 General Large Sample Methods 631
15.1 Introduction 631
15.2 Permutation and Randomization Tests 632
15.2.1 The Basic Construction 632
15.2.2 Asymptotic Results 636
15.3 Basic Large Sample Approximations 643
15.3.1 Pivotal Method 644
15.3.2 Asymptotic Pivotal Method 646
15.3.3 Asymptotic Approximation 647
15.4 Bootstrap Sampling Distributions 648
15.4.1 Introduction and Consistency 648
15.4.2 The Nonparametric Mean 653
15.4.3 Further Examples 655
15.4.4 Stepdown Multiple Testing 658
15.5 Higher Order Asymptotic Comparisons 661
15.6 Hypothesis Testing 668
15.7 Subsampling 673
15.7.1 The Basic Theorem in the I.I.D. Case 674
15.7.2 Comparison with the Bootstrap 677
15.7.3 Hypothesis Testing 680
15.8 Problems 682
15.9 Notes 690
A Auxiliary Results 692
A.I Equivalence Relations; Groups 692
xiv Contents
A.2 Convergence of Functions; Metric Spaces 693
A.3 Banach and Hilbert Spaces 696
A.4 Dominated Families of Distributions 698
A.5 The Weak Compactness Theorem 700
References 702
Author Index 757
Subject Index 767 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Lehmann, Erich L. 1917-2009 Romano, Joseph P. 1960- |
| author_GND | (DE-588)120765241 (DE-588)12146878X |
| author_facet | Lehmann, Erich L. 1917-2009 Romano, Joseph P. 1960- |
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| author_sort | Lehmann, Erich L. 1917-2009 |
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| building | Verbundindex |
| bvnumber | BV022374841 |
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| callnumber-subject | QA - Mathematics |
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| classification_tum | MAT 000 MAT 625f |
| collection | ZDB-2-SMA |
| ctrlnum | (OCoLC)249833198 (DE-599)BVBBV022374841 |
| dewey-full | 519.56 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.56 |
| dewey-search | 519.56 |
| dewey-sort | 3519.56 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| doi_str_mv | 10.1007/0-387-27605-X |
| edition | 3. ed. |
| format | Electronic eBook |
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| id | DE-604.BV022374841 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:08:47Z |
| indexdate | 2024-07-09T20:56:15Z |
| institution | BVB |
| isbn | 9780387276052 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015583932 |
| oclc_num | 249833198 |
| open_access_boolean | |
| owner | DE-634 DE-355 DE-BY-UBR DE-384 DE-703 DE-91 DE-BY-TUM DE-83 |
| owner_facet | DE-634 DE-355 DE-BY-UBR DE-384 DE-703 DE-91 DE-BY-TUM DE-83 |
| physical | 1 Online-Ressource (XIV, 784 S.) graph. Darst. |
| psigel | ZDB-2-SMA |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer texts in statistics |
| spelling | Lehmann, Erich L. 1917-2009 Verfasser (DE-588)120765241 aut Testing statistical hypotheses E. L. Lehmann ; Joseph P. Romano 3. ed. New York, NY Springer 2005 1 Online-Ressource (XIV, 784 S.) graph. Darst. txt rdacontent c rdamedia cr rdacarrier Springer texts in statistics Includes bibliographical references and index aStatistical hypothesis testing Statistik (DE-588)4056995-0 gnd rswk-swf Statistischer Test (DE-588)4077852-6 gnd rswk-swf Statistische Hypothese (DE-588)4182959-1 gnd rswk-swf Statistischer Test (DE-588)4077852-6 s DE-604 Statistische Hypothese (DE-588)4182959-1 s 1\p DE-604 Statistik (DE-588)4056995-0 s 2\p DE-604 Romano, Joseph P. 1960- Verfasser (DE-588)12146878X aut Erscheint auch als Druck-Ausgabe 0-387-98864-5 (DE-604)BV019769340 https://doi.org/10.1007/0-387-27605-X Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015583932&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lehmann, Erich L. 1917-2009 Romano, Joseph P. 1960- Testing statistical hypotheses aStatistical hypothesis testing Statistik (DE-588)4056995-0 gnd Statistischer Test (DE-588)4077852-6 gnd Statistische Hypothese (DE-588)4182959-1 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4077852-6 (DE-588)4182959-1 |
| title | Testing statistical hypotheses |
| title_auth | Testing statistical hypotheses |
| title_exact_search | Testing statistical hypotheses |
| title_exact_search_txtP | Testing statistical hypotheses |
| title_full | Testing statistical hypotheses E. L. Lehmann ; Joseph P. Romano |
| title_fullStr | Testing statistical hypotheses E. L. Lehmann ; Joseph P. Romano |
| title_full_unstemmed | Testing statistical hypotheses E. L. Lehmann ; Joseph P. Romano |
| title_short | Testing statistical hypotheses |
| title_sort | testing statistical hypotheses |
| topic | aStatistical hypothesis testing Statistik (DE-588)4056995-0 gnd Statistischer Test (DE-588)4077852-6 gnd Statistische Hypothese (DE-588)4182959-1 gnd |
| topic_facet | aStatistical hypothesis testing Statistik Statistischer Test Statistische Hypothese |
| url | https://doi.org/10.1007/0-387-27605-X http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015583932&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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