Concepts of nonparametric theory:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1981
|
| Series: | Springer series in statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 445 - 453 |
| Physical Description: | XVI, 462 S. graph. Darst. |
| ISBN: | 0387905820 3540905820 |
Staff View
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| 245 | 1 | 0 | |a Concepts of nonparametric theory |c John W. Pratt ; Jean D. Gibbons |
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Record in the Search Index
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Contents
CHAPTER 1
Concepts of Statistical Inference and the Binomial Distribution 1
1 Introduction 1
2 Probability Distributions 2
3 Estimators and their Properties 6
3.1 Unbiasedness and Variance 7
3.2 Consistency 8
3.3 Sufficiency 8
3.4 Minimum Variance 12
4 Hypothesis Testing 14
4.1 Tests and their Interpretation 14
4.2 Errors 17
4.3 One-Tailed Binomial Tests 22
4.4 P-values 23
4.5 Two-Tailed Test Procedures and P-values 28
4.6 Other Conclusions in Two-Tailed Tests 32
5 Randomized Test Procedures 34
5.1 Introduction: Motivation and Examples 34
5.2 Randomized Tests: Definitions 37
5.3 Nonrandomized Tests Equivalent to Randomized Tests 38
5.4 Usefulness of Randomized Tests in Theory and Practice 39
5.5 *Randomized P-values 40
6 Confidence Regions 41
6.1 Definition and Construction in the Binomial Case 41
6.2 Definition of Confidence Regions and Relationship to Tests in the
General Case 45
6.3 Interpretation of Confidence Regions 46
6.4 True Confidence Level 48
xi
xii Contents
6.5 Including False Values and the Size of Confidence Regions 49
6.6 *Randomized Confidence Regions 51
7 Properties of One-Tailed Procedures 52
7.1 Uniformly Most Powerful One-Tailed Tests 52
7.2 *Admissibility and Completeness of One-Tailed Tests 53
7.3 Confidence Procedures 54
7.4 *Proofs 55
8 Choice of Two-Tailed Procedures and their Properties 58
8.1 Test Procedures 58
8.2 Confidence Procedures 61
8.3 *Completeness and Admissibility of Two-Conclusion
Two-Tailed Tests 63
8.4 ""Completeness and Admissibility of Three-Conclusion
Two-Tailed Tests 64
9 Appendices to Chapter 1 65
A Limits of the Binomial Distribution 66
A.I Ordinary Poisson Approximation and Limit 66
A.2 Ordinary Normal Approximation and Limit 67
B Convergence in Distribution and Asymptotic Distributions 69
B.I Convergence of Frequency Functions and Densities 69
B.2 Convergence in Distribution 72
B.3 Two Central Limit Theorems 73
Problems 74
CHAPTER 2
One-Sample and Paired-Sample Inferences Based on the Binomial
Distribution 82
1 Introduction 82
2 Quantile Values 83
3 The One-Sample Sign Test for Quantile Values 85
3.1 Test Procedures 85
3.2 "Optimum" Properties 88
3.3 *Proofs 91
4 Confidence Procedures Based on the Sign Test 92
5 Interpolation between Attainable Levels 96
6 The Sign Test with Zero Differences 97
6.1 Discussion of Procedures 97
6.2 Conditional Properties of Conditional Sign Tests 99
6.3 Unconditional Properties of Conditional Sign Tests 101
6.4 *Proof for One-Sided Alternatives 101
6.5 *Proof for Two-Sided Alternatives 102
7 Paired Observations 104
8 Comparing Proportions using Paired Observations 106
8.1 Test Procedure 108
8.2 Alternative Presentations 109
8.3 Example 110
8.4 Interpretation of the Test Results 112
8.5 Properties of the Test 114
Contents xiii
8.6 Other Inferences 114
8.7 Cox Model 116
9 Tolerance Regions 118
9.1 Definition 118
9.2 Practical Uses 119
9.3 Construction of Tolerance Regions: Wilks' Method 121
9.4 Tolerance Regions for Description 124
9.5 Tolerance Regions for Prediction 126
9.6 More General Construction Procedures 127
Problems 130
CHAPTER 3
One-Sample and Paired-Sample Inferences Based on Signed Ranks 145
1 Introduction 145
2 The Symmetry Assumption or Hypothesis 146
3 The Wilcoxon Signed-Rank Test 147
3.1 Test Procedure and Exact Null Distribution Theory 147
3.2 Asymptotic Null Distribution Theory 150
3.3 Large Sample Power 151
3.4 Consistency 153
3.5 Weakening the Assumptions 155
4 Confidence Procedures Based on the Wilcoxon Signed-Rank
Test 157
5 A Modified Wilcoxon Procedure 158
6 Zeros and Ties 160
6.1 Introduction 160
6.2 Obtaining the Signed Ranks 162
6.3 Test Procedures 163
6.4 Warnings and Anomalies: Examples 167
6.5 Comparison of Procedures 170
7 Other Signed-Rank Procedures 171
7.1 Sums of Signed Constants 172
7.2 Signed Ranks and Walsh Averages 173
7.3 Confidence Bounds Corresponding to Signed-Rank Tests 174
7.4 Procedures Involving a Small Number of Walsh Averages 175
8 Invariance and Signed-Rank Procedures 177
8.1 Permutation Invariance 177
8.2 Invariance under Increasing, Odd Transformations 179
9 Locally Most Powerful Signed-Rank Tests 181
Problems 185
CHAPTER 4
One-Sample and Paired-Sample Inferences Based on the Method of
Randomization 203
1 Introduction 203
2 Randomization Procedures Based on the Sample Mean and
Equivalent Criteria 205
xiv Contents
2.1 Tests 205
2.2 Weakening the Assumptions 208
2.3 Related Confidence Procedures 209
2.4 Properties of the Exact Randomization Distribution 210
2.5 Modifications and Approximations to the Exact Randomization
Procedure 210
3 The General Class of One-Sample Randomization Tests 216
3.1 Definition 216
3.2 Properties 217
4 Most Powerful Randomization Tests 222
4.1 General Case 222
4.2 One-Sided Normal Alternatives 223
4.3 Two-Sided Normal Alternatives 223
5 Obervation-Randomization versus Rank-Randomization Tests 224
Problems 226
CHAPTER 5
Two-Sample Rank Procedures for Location 231
1 Introduction 231
2 The Shift Assumption 232
3 The Median Test, Other Two-Sample Sign Tests, and Related
Confidence Procedures 234
3.1 Reduction of Data to a 2 x 2 Table 234
3.2 Fisher's Exact Test for 2 x 2 Tables 238
3.3 Ties 241
3.4 Corresponding Confidence Procedures 242
3.5 Power 243
3.6 Consistency 245
3.7 "Optimum" Properties 246
3.8 Weakening the Assumptions 247
4 Procedures Based on Sums of Ranks 249
4.1 The Rank Sum Test Procedure 249
4.2 Null Distribution of the Rank Sum Statistics 252
4.3 Corresponding Confidence Procedures 253
4.4 Approximate Power 255
4.5 Consistency 257
4.6 Weakening the Assumptions 257
4.7 Ties 258
4.8 Point and Confidence Interval Estimation of P(X Y) 263
5 Procedures Based on Sums of Scores 265
6 Two-Sample Rank Tests and the Y - X Differences 269
7 Invariance and Two-Sample Rank Procedures 269
8 Locally Most Powerful Rank Tests 272
8.1 Most Powerful and Locally Most Powerful Rank Tests Against
Given Alternatives 272
8.2 The Class of Locally Most Powerful Rank Tests 277
Problems 279
Contents XV
CHAPTER 6
Two-Sample Inferences Based on the Method of Randomization 296
1 Introduction 296
2 Randomization Procedures Based on the Difference Between Sample
Means and Equivalent Criteria 297
2.1 Tests 297
2.2 Weakening the Assumptions 298
2.3 Related Confidence Procedures 300
2.4 Properties of the Exact Randomization Distribution 301
2.5 Approximations to the Exact Randomization Distribution 302
3 The Class of Two-Sample Randomization Tests 305
3.1 Definition 305
3.2 Properties 307
4 Most Powerful Randomization Tests 310
4.1 General Case 310
4.2 One-Sided Normal Alternatives 311
4.3 Two-Sided Normal Alternatives 312
Problems 314
CHAPTER 7
Kolmogorov-Smirnov Two-Sample Tests 318
1 Introduction 318
2 Empirical Distribution Function 319
3 Two-Sample Kolmogorov-Smirnov Statistics 320
4 Null Distribution Theory 322
4.1 An Algorithm for the Exact Null Distribution 323
4.2 Relation Between One-Tailed and Two-Tailed Procedures 325
4.3 Exact Formulas for Equal Sample Sizes 325
4.4 Asymptotic Null Distributions 328
5 Ties 330
6 Performance 331
7 One-Sample Kolmogorov-Smirnov Statistics 334
Problems 336
CHAPTER 8
Asymptotic Relative Efficiency 345
1 Introduction 345
2 Asymptotic Behavior of Tests: Heuristic Discussion 347
2.1 Asymptotic Power of a Test 347
2.2 Nuisance Parameters 351
2.3 Asymptotic Relative Efficiency of Two Tests 353
3 Asymptotic Behavior of Point Estimators: Heuristic Discussion 355
3.1 Estimators of the Same Quantity 355
3.2 Relation of Estimators and Tests 357
3.3 *Estimators of Different Quantities 360
xvi Contents
4 Asymptotic Behavior of Confidence Bounds 362
5 Example 364
6 *Definitions of Asymptotic Relative Efficiency 370
6.1 Introduction 370
6.2 Tests 371
6.3 Estimators 373
6.4 Confidence Bounds 374
6.5 Confidence Intervals 376
6.6 Summary 376
7 Pitman's Formula 377
8 Asymptotic Relative Efficiencies of One-Sample Procedures for
Shift Families 378
8.1 The Shift Model 379
8.2 Asymptotic Relative Efficiencies for Specific Shift Families 379
8.3 Bounds on Asymptotic Relative Efficiencies for Shift Families 388
8.4 *Asymptotically Efficient Signed-Rank Procedures 392
9 Asymptotic Relative Efficiency of Procedures for Matched Pairs 394
9.1 Assumptions 394
9.2 Bounds for Asymptotic Efficiencies for Matched Pairs 397
10 Asymptotic Relative Efficiency of Two-Sample Procedures for
Shift Families 398
11 Asymptotic Efficiency of Kolmogorov-Smirnov Procedures 401
11.1 One-Sided Tests 402
11.2 Two-Sided Tests 409
Problems 412
Tables 425
Table A Cumulative Standard Normal Distribution 426
Table B Cumulative Binomial Distribution 428
Table C Binomial Confidence Limits 431
Table D Cumulative Probabilities for Wilcoxon Signed-Rank Statistic 433
Table E Cumulative Probabilities for Hypergeometric Distribution 435
Table F Cumulative Probabilities for Wilcoxon Rank Sum Statistic 437
Table G Kolmogorov-Smirnov Two-Sample Statistic 443
Bibliography 445
T A 455
Index |
| any_adam_object | 1 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| id | DE-604.BV001018357 |
| illustrated | Illustrated |
| indexdate | 2024-10-15T10:03:04Z |
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| isbn | 0387905820 3540905820 |
| language | English |
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| physical | XVI, 462 S. graph. Darst. |
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| spelling | Pratt, John W. 1931- Verfasser (DE-588)133010201 aut Concepts of nonparametric theory John W. Pratt ; Jean D. Gibbons New York [u.a.] Springer 1981 XVI, 462 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Literaturverz. S. 445 - 453 Statistique non-paramétrique Nonparametric statistics Nichtparametrisches Verfahren (DE-588)4339273-8 gnd rswk-swf Nichtparametrische Statistik (DE-588)4226777-8 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf Nichtparametrische Statistik (DE-588)4226777-8 s DE-604 Statistik (DE-588)4056995-0 s Nichtparametrisches Verfahren (DE-588)4339273-8 s Gibbons, Jean Dickinson 1938- Verfasser (DE-588)132991373 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000615830&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pratt, John W. 1931- Gibbons, Jean Dickinson 1938- Concepts of nonparametric theory Statistique non-paramétrique Nonparametric statistics Nichtparametrisches Verfahren (DE-588)4339273-8 gnd Nichtparametrische Statistik (DE-588)4226777-8 gnd Statistik (DE-588)4056995-0 gnd |
| subject_GND | (DE-588)4339273-8 (DE-588)4226777-8 (DE-588)4056995-0 |
| title | Concepts of nonparametric theory |
| title_auth | Concepts of nonparametric theory |
| title_exact_search | Concepts of nonparametric theory |
| title_full | Concepts of nonparametric theory John W. Pratt ; Jean D. Gibbons |
| title_fullStr | Concepts of nonparametric theory John W. Pratt ; Jean D. Gibbons |
| title_full_unstemmed | Concepts of nonparametric theory John W. Pratt ; Jean D. Gibbons |
| title_short | Concepts of nonparametric theory |
| title_sort | concepts of nonparametric theory |
| topic | Statistique non-paramétrique Nonparametric statistics Nichtparametrisches Verfahren (DE-588)4339273-8 gnd Nichtparametrische Statistik (DE-588)4226777-8 gnd Statistik (DE-588)4056995-0 gnd |
| topic_facet | Statistique non-paramétrique Nonparametric statistics Nichtparametrisches Verfahren Nichtparametrische Statistik Statistik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000615830&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT prattjohnw conceptsofnonparametrictheory AT gibbonsjeandickinson conceptsofnonparametrictheory |