Theory of rank tests:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
San Diego [u.a.]
Acad. Press
1999
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| Edition: | 2. ed. |
| Series: | Probability and mathematical statistics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 435 S. |
| ISBN: | 0126423504 |
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Record in the Search Index
| _version_ | 1804127269817417728 |
|---|---|
| adam_text | Titel: Theory of rank tests
Autor: Hájek, Jaroslav
Jahr: 1999
Contents
Preface to the Second Edition v
Preface to the First Edition vii
1 Introduction and coverage 1
1.1 THE BACKGROUND......................... 1
1.2 ORGANIZATION OF THE PRESENT TREATISE........ 5
2 Preliminaries 10
2.1 BASIC NOTATION.......................... 10
2.1.1 Probability space and observations............... 10
2.1.2 Statistics, cr-fields and A-fields................. 11
2.1.3 Martingales and related sequences............... 13
2.2 FAMILIES OF ONE-DIMENSIONAL DENSITIES......... 13
2.2.1 Absolutely continuous densities................ 14
2.2.2 Strongly unimodal densities..............• • • • 14
2.2.3 Densities with finite Fisher information............ 16
2.2.4 The (^-functions......................... 17
2.2.5 Types of densities and the measurement of their distance. . 21
2.3 TESTING HYPOTHESES ...................... 22
2.3.1 Statement of the problem.................... 22
2.3.2 The envelope power function.................. 25
2.3.3 Most powerful test........................ 26
2.3.4 The Neyman-Pearson lemma.................. 27
2.3.5 Least favourable densities.................... 28
2.4 AUXILIARY RESULTS FOR NORMAL SAMPLES........ 29
2.4.1 ^-distribution for correlated normal random variables. . . 29
2.4.2 Most powerful tests....................... 30
Problems and complements to Chapter 2 ................. 31
Contents
Elementary theory of rank tests 35
3.1 RANKS AND ORDER STATISTICS.................35
3.1.1 Ranks and order statistics................... 35
3.1.2 Hypotheses Ho and H» (randomness)............. 37
3.1.3 Hypothesis Hi (symmetry)................... 39
3.1.4 Hypothesis H2 (independence)................. 40
3.1.5 Hypothesis H3 (random blocks)................ 41
3.2 PERMUTATION, INVARIANT, AND RANK TESTS....... 41
3.2.1 Permutation tests........................ 41
3.2.2 Tests invariant under changes of location and scale..... 45
3.2.3 Rank tests............................ 52
3.2.4 Lehmann s alternatives..................... 54
3.3 EXPECTATIONS AND VARIANCES OF LINEAR RANK
STATISTICS.............................. 57
3.3.1 Linear rank statistics for Ho..................57
3.3.2 Linear rank statistics for Hi, H2 and Hz- Antiranks.....62
3.4 LOCALLY MOST POWERFUL RANK TESTS..........64
3.4.1 Definition of locally most powerful tests............ 64
3.4.2 A convergence theorem..................... 64
3.4.3 Scores............................... 65
3.4.4 Ho against two samples differing in location......... 67
3.4.5 Ho against two samples differing in scale........... 69
3.4.6 Ho against regression in location................ 69
3.4.7 Ho against regression in scale.................. 70
3.4.8 Ho against a general alternative................ 70
3.4.9 Hi against the location shift.................. 74
3.4.10 Hi against two samples differing in scale........... 75
3.4.11 H2 against dependence..................... 76
3.5 STATISTICAL FUNCTIONALS................... 79
3.5.1 [/-statistics and V-statistics.................. 80
3.5.2 Differentiable statistical functional.............. 83
3.5.3 Generalized statistical functionals............... 84
Problems and complements to Chapter 3 ................. 85
Selected rank tests 94
4.1 TWO-SAMPLE TESTS OF LOCATION..............95
4.1.1 Linear rank tests.........................95
4.1.2 Tests based on exceeding observations.............98
4.1.3 Tests of Kolmogorov-Smirnov types..............99
4.2 TWO-SAMPLE TESTS OF SCALE.................103
4.2.1 Linear rank tests.........................103
4.2.2 Tests based on exceeding observations.............107
4.2.3 Tests of Kolmogorov-Smirnov types..............108
4.3 REGRESSION.............................108
4.3.1 Linear rank tests.........................109
4.3.2 The £-test............................110
4.3.3 Tests of Kolmogorov-Smirnov types..............Ill
4.4 THREE OR MORE SAMPLES....................112
Contents
4.4.1 Rank tests of x2-types.....................112
4.4.2 Other rank tests and some general ideas...........114
4.5 TESTS OF SYMMETRY.......................117
4.5.1 Linear rank tests.........................117
4.5.2 Kolmogorov-type tests.....................119
4.6 TESTS OF INDEPENDENCE....................121
4.6.1 Linear rank tests.........................121
4.6.2 Other rank tests.........................126
4.6.3 Kolmogorov-type tests.....................126
4.7 RANDOM BLOCKS..........................128
4.7.1 Rank tests of x2-types.....................128
4.8 TREATMENT OF TIES .......................130
4.8.1 Randomization, average statistics and scores, mid-ranks. . . 131
4.8.2 Other methods of dealing with ties..............135
4.9 RANK TESTS FOR CENSORED DATA..............136
4.9.1 Type I censoring.........................136
4.9.2 Type II censoring........................139
4.9.3 Progressive censoring......................141
4.9.4 Interval censoring........................144
4.9.5 Random censoring.......................148
4.10 MULTIVARIATE RANK TESTS...................152
4.10.1 Bivariate sign tests.......................152
4.10.2 Multivariate signed-rank tests.................153
4.10.3 Multivariate linear rank statistics...............155
4.10.4 LMPR property for multivariate tests............159
Problems and complements to Chapter 4 .................160
Computation of null exact distributions 165
5.1 DIRECT USE OF DISTRIBUTION OF RANKS..........165
5.1.1 Tests for regression, independence and random blocks. . . . 165
5.1.2 Two-sample tests........................167
5.1.3 Tests for three or more samples................167
5.1.4 Tests of symmetry.......................168
5.2 EXPLICIT FORMULAS FOR DISTRIBUTIONS .........168
5.2.1 Statistics using the scores 0, , 1................168
5.2.2 The Haga test and the Ê-test statistics............170
5.2.3 Kolmogorov-Smirnov statistics for two samples of equal sizes. 170
5.3 RECURRENCE FORMULAS FOR DISTRIBUTIONS.......173
5.3.1 The Wilcoxon and the Kendall test statistics.........173
5.3.2 Statistics of Kolmogorov-Smirnov types for two samples of
unequal sizes...........................174
5.3.3 The Kruskal-Wallis and the Friedman test statistics.....176
5.4 IMPROVEMENTS OF LIMITING DISTRIBUTIONS.......177
5.4.1 Corrections for continuity.....................177
5.4.2 Interpolation for numerical computations...........178
5.4.3 Transformations of test statistics and the use of other lim-
iting distributions........................179
Problems and complements to Chapter 5 .................179
Contents
Limiting null distributions 183
6.1 SIMPLE LINEAR RANK STATISTICS...............183
6.1.1 Convergence in indexed sets of statistics...........183
6.1.2 A special central limit theorem.................184
6.1.3 A convergence theorem.....................186
6.1.4 Further preliminaries......................186
6.1.5 Locally optimum rank-test statistics for Ho..........190
6.1.6 General simple linear rank statistics for Ho..........194
6.1.7 Rank statistics for Hi......................197
6.1.8 Simple linear rank statistics for H-¿..............199
6.1.9 Martingale characterizations..................200
6.2 RANK STATISTICS OF X2-TYPES.................203
6.2.1 Convergence in distribution for random vectors........203
6.2.2 Statistics for testing Ho against k samples differing in lo-
cation or scale..........................205
6.2.3 Statistics for H3 against differences within blocks......208
6.3 STATISTICS OF KOLMOGOROV-SMIRNOV TYPES......209
6.3.1 Probability distributions in the space of continuous functions.209
6.3.2 Convergence in distribution in C[0,1].............215
6.3.3 Weak convergence in the Skorokhod metric..........217
6.3.4 The Brownian bridge......................219
6.3.5 Kolmogorov s inequality for dependent summands......222
6.3.6 The basic convergence theorem.................224
6.3.7 The Kolmogorov-Smirnov statistics..............227
6.3.8 The Rényi statistics.......................228
6.3.9 The Cramér-von Mises statistics................230
6.4 FUNCTIONAL CENTRAL LIMIT THEOREMS..........230
6.4.1 Weak invariance principles for martingales..........231
6.4.2 Almost sure invariance principles for martingales.......232
6.4.3 Invariance principles for empirical processes.........233
6.4.4 Invariance principles for rank statistics............234
Problems and complements to Chapter 6 .................240
Limiting non-null distributions 249
7.1 CONTIGUITY.............................249
7.1.1 Asymptotic methods. Contiguity................249
7.1.2 LeCam s first lemma......................251
7.1.3 LeCam s second lemma.....................253
7.1.4 LeCam s third lemma......................257
7.1.5 LAQ family and the convolution theorem...........260
7.1.6 Contiguity and Hellinger distance...............261
7.2 SIMPLE LINEAR RANK STATISTICS...............262
7.2.1 Location alternatives for Ho..................262
7.2.2 Scale alternatives for Ho....................265
7.2.3 Rank statistics for Ho against two samples..........267
7.2.4 Rank statistics for Ho against regression...........269
7.2.5 Rank statistics for Hi......................274
7.2.6 Rank statistics for H2......................275
Contents
7.3 FAMILIES OF SIMPLE LINEAR RANK STATISTICS......276
7.3.1 Finite families (vectors).....................276
7.3.2 Continuous families (processes).................279
7.4 ASYMPTOTIC POWER.......................281
7.4.1 One-sided S-tests........................281
7.4.2 Two-sided S-tests........................282
7.4.3 Q-tests..............................283
7.4.4 Tests of Kolmogorov-Smirnov types..............283
7.4.5 Local behaviour of the asymptotic power (one-sided tests). 284
7.4.6 Local behaviour of asymptotic power (multi-sided tests). . . 287
7.4.7 Progressive censoring......................288
7.5 NON-CONTIGUOUS ALTERNATIVES...............289
7.5.1 A historical note.........................289
7.5.2 Variance inequality for linear rank statistics.........290
7.5.3 The Hájek projection approximation..............291
7.5.4 Asymptotic normality of linear rank statistics........292
7.5.5 Integration with contiguity based approaches.........293
Problems and complements to Chapter 7 .................294
Asymptotic optimality and efficiency 299
8.1 ASYMPTOTICALLY OPTIMUM TESTS..............299
8.1.1 Introduction...........................299
8.1.2 Asymptotic sufficiency of the vector of ranks.........302
8.1.3 Asymptotically optimum one-sided tests...........307
8.1.4 Asymptotically optimum multi-sided tests..........310
8.2 ASYMPTOTIC EFFICIENCY OF TESTS.............316
8.2.1 One-sided tests..........................316
8.2.2 Multi-sided tests.........................320
8.2.3 Local asymptotic efficiency...................321
Bibliographical notes..........................323
8.3 BAHADUR EFFICIENCY......................325
8.3.1 Exact Bahadur efficiency....................325
8.3.2 Kullback-Leibler information..................330
8.3.3 Best exact slope of two-sample rank statistics........332
8.3.4 Approximate Bahadur efficiency................ 336
8.3.5 Relation between Bahadur efficiency and Pitman efficiency. 336
8.4 HODGES-LEHMANN DEFICIENCY................336
8.4.1 Asymptotic expansions for distributions of rank statistics. . 337
8.4.2 Hodges-Lehmann s concept of deficiency............340
8.5 ADAPTIVE RANK TESTS......................341
8.5.1 Restrictive adaptive tests....................342
8.5.2 Non-restrictive adaptive tests..................345
Problems and complements to Chapter 8 .................352
Rank estimates and asymptotic linearity 358
9.1 ñ-ESTIMATES OF LOCATION AND REGRESSION.......358
9.1.1 .R-estimates of shift between two samples...........358
9.1.2 iï-estimates of shift of one sample...............361
xiv Contents
9.1.3 K-estimates of regression parameters.............362
9.2 ASYMPTOTIC LINEARITY OF RANK STATISTICS IN
REGRESSION PARAMETERS....................366
9.2.1 Rank statistics, one-dimensional regression..........366
9.2.2 Rank statistics, multidimensional regression.........377
9.2.3 Signed-rank statistics .....................378
9.3 RANK ESTIMATION OF REGRESSION PARAMETERS .... 378
9.3.1 Basic methods..........................378
9.3.2 Linearized rank estimates...................381
Problems and complements to Chapter 9 .................382
10 Miscellaneous topics in regression rank tests 383
10.1 ALIGNED RANK TESTS.......................383
10.1.1 Aligned rank tests for two-way layouts............384
10.1.2 Aligned rank statistics for subhypothesis testing.......389
10.2 REGRESSION RANK SCORES...................394
10.2.1 The duality of RQ and RRS..................395
10.2.2 RRS estimates and connection to location model.......397
10.3 RANK VERSUS OTHER ROBUST PROCEDURES........399
Problems and complements to Chapter 10.................404
Bibliography 407
Subject index 425
Author index 429
Index of mathematical symbols 433
Titles in this series 437
|
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| indexdate | 2024-07-09T18:30:40Z |
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| language | English |
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| spelling | Hajek, Jaroslav Verfasser aut Theory of rank tests Jaroslav Hájek ; Zbyněk Šidák ; Pranab K. Sen 2. ed. San Diego [u.a.] Acad. Press 1999 XIV, 435 S. txt rdacontent n rdamedia nc rdacarrier Probability and mathematical statistics Asymptotische analyse gtt Hypothesetoetsing gtt Rang et sélection (statistique) ram Statistique mathématique ram Tests d'hypothèses (statistique) ram inférence paramétrique inriac test non paramétrique inriac test rang inriac test statistique inriac Ranking and selection (Statistics) Statistical hypothesis testing Rangtest (DE-588)4210732-5 gnd rswk-swf Rangtest (DE-588)4210732-5 s DE-604 Šidák, Zbyněk Verfasser aut Sen, Pranab Kumar 1937- Verfasser (DE-588)124306942 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008569414&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hajek, Jaroslav Šidák, Zbyněk Sen, Pranab Kumar 1937- Theory of rank tests Asymptotische analyse gtt Hypothesetoetsing gtt Rang et sélection (statistique) ram Statistique mathématique ram Tests d'hypothèses (statistique) ram inférence paramétrique inriac test non paramétrique inriac test rang inriac test statistique inriac Ranking and selection (Statistics) Statistical hypothesis testing Rangtest (DE-588)4210732-5 gnd |
| subject_GND | (DE-588)4210732-5 |
| title | Theory of rank tests |
| title_auth | Theory of rank tests |
| title_exact_search | Theory of rank tests |
| title_full | Theory of rank tests Jaroslav Hájek ; Zbyněk Šidák ; Pranab K. Sen |
| title_fullStr | Theory of rank tests Jaroslav Hájek ; Zbyněk Šidák ; Pranab K. Sen |
| title_full_unstemmed | Theory of rank tests Jaroslav Hájek ; Zbyněk Šidák ; Pranab K. Sen |
| title_short | Theory of rank tests |
| title_sort | theory of rank tests |
| topic | Asymptotische analyse gtt Hypothesetoetsing gtt Rang et sélection (statistique) ram Statistique mathématique ram Tests d'hypothèses (statistique) ram inférence paramétrique inriac test non paramétrique inriac test rang inriac test statistique inriac Ranking and selection (Statistics) Statistical hypothesis testing Rangtest (DE-588)4210732-5 gnd |
| topic_facet | Asymptotische analyse Hypothesetoetsing Rang et sélection (statistique) Statistique mathématique Tests d'hypothèses (statistique) inférence paramétrique test non paramétrique test rang test statistique Ranking and selection (Statistics) Statistical hypothesis testing Rangtest |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008569414&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT hajekjaroslav theoryofranktests AT sidakzbynek theoryofranktests AT senpranabkumar theoryofranktests |