Comparing distributions:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2010
|
| Series: | Springer series in statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 339 - 349 |
| Physical Description: | XVIII, 353 S. graph. Darst. |
| ISBN: | 9780387927091 |
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| 100 | 1 | |a Thas, Olivier |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Comparing distributions |c Olivier Thas |
| 264 | 1 | |a New York, NY |b Springer |c 2010 | |
| 300 | |a XVIII, 353 S. |b graph. Darst. | ||
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| 500 | |a Literaturverz. S. 339 - 349 | ||
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| adam_text | Contents
Preface
.......................................................
vii
Part I One-Sample Problems
1
Introduction
.............................................. 3
1.1
The History of the One-Sample GOF Problem
............. 3
1.2
Example
Datasets
...................................... 4
1.2.1
Pseudo-Random Generator Data
................... 4
1.2.2
PCB
Concentration Data
.......................... 5
1.2.3
Pulse Rate Data
................................. 5
1.2.4
Cultivare
Data
................................... 6
1.3
The Pearson Chi-Squared Test
........................... 8
1.3.1
Pearson Chi-Squared Test for the Multinomial
Distribution
..................................... 8
1.3.2
Generalisations of the Pearson
χ2
Test
.............. 13
1.3.3
A Note on the Nuisance Parameter Estimation
....... 14
1.4
Pearson X2 Tests for Continuous Distributions
............. 15
2
Preliminaries (Building Blocks)
........................... 19
2.1
The Empirical Distribution Function
...................... 19
2.1.1
Definition and Construction
....................... 19
2.1.2
Rationale for Using the EDF
....................... 21
2.2
Empirical Processes
..................................... 22
2.2.1
Definition
....................................... 22
2.2.2
Weak Convergence
............................... 23
2.2.3
Kac-Siegert Decomposition of Gausian Processes
..... 24
2.3
The Quantile Function and the Quantile Process
........... 27
2.3.1
The Quantile Function and Its Estimator
............ 27
2.3.2
The Quantile Process
............................. 28
2.4
Comparison Distribution
................................ 29
2.5
Hubert Spaces
......................................... 30
Contents
2.6
Orthonormal
Functions
.................................. 33
2.6.1
The Fourier Basis
................................ 33
2.6.2
Orthonormal
Polynomials
......................... 33
2.7
Parameter Estimation
................................... 34
2.7.1
Locally Asymptotically Linear Estimators
........... 34
2.7.2
Method of Moments Estimators
.................... 35
2.7.3
Efficiency and Semiparametric Inference
............. 36
2.8
Nonparametric Density Estimation
....................... 37
2.8.1
Introduction
..................................... 37
2.8.2
Orthogonal Series Estimators
...................... 39
2.8.3
Kernel Density Estimation
......................... 42
2.8.4
Regression-Based Density Estimation
............... 42
2.9
Hypothesis Testing
..................................... 42
2.9.1
General Construction of a Hypothesis Test
.......... 43
2.9.2
Optimality Criteria
............................... 44
2.9.3
The Neyman-Pearson Lemma
...................... 47
Graphical Tools
........................................... 49
3.1
Histograms and Box Plots
............................... 49
3.1.1
The Histogram
................................... 49
3.1.2
The Box Plot
.................................... 52
3.2
Probability Plots and Comparison Distribution
............. 56
3.2.1
Population Probability Plots
....................... 56
3.2.2
PP and QQ plots
................................. 57
3.3
Comparison Distribution
................................ 62
3.3.1
Population Comparison Distributions
............... 62
3.3.2
Empirical Comparison Distributions
................ 68
3.3.3
Comparison Distribution for Discrete Data
.......... 73
Smooth Tests
............................................. 77
4.1
Smooth Models
........................................ 77
4.1.1
Construction of the Smooth Model
................. 77
4.2
Smooth Tests
.......................................... 82
4.2.1
Simple Null Hypotheses
........................... 82
4.2.2
Composite Null Hypotheses
........................ 88
4.3
Adaptive Smooth Tests
................................. 95
4.3.1
Consistency, Dilution Effects and Order Selection
..... 95
4.3.2
Order Selection Within a Finite Horizon
............ 98
4.3.3
Order Selection Within an Infinite Horizon
.......... 102
4.3.4
Subset Selection Within a Finite Horizon
............ 103
4.3.5
Improved Density Estimates
....................... 107
4.4
Smooth Tests for Discrete Distributions
................... 108
4.4.1
Introduction
..................................... 108
4.4.2
The Simple Null Hypothesis Case
.................. 108
4.4.3
The Composite Null Hypothesis Case
............... 109
Contents xv
4.5
A Semiparametric
Framework............................111
4.5.1 The Semiparametric
Hypotheses ...................Ill
4.5.2 Semiparametric Tests.............................112
4.5.3
A Distance Function
..............................114
4.5.4 Interpretation
and Estimation of the Nuisance
Parameter
.......................................114
4.5.5
The Quadratic Inference Function
..................115
4.5.6
Relation with the Empirically Rescaled Smooth Tests
. 116
4.6
Example
..............................................117
4.7
Some Practical Guidelines for Smooth Tests
...............121
5
Methods Based on the Empirical Distribution Function
... 123
5.1
The Kolmogorov-Smirnov Test
...........................123
5.1.1
Definition
.......................................123
5.1.2
Null Distribution
.................................125
5.1.3
Presence of Nuisance Parameters
...................127
5.2
Tests as Integrals of Empirical Processes
..................129
5.2.1
The Anderson-Darling Statistics
...................129
5.2.2
Principal Components Decomposition of the Test
Statistic
.........................................130
5.2.3
Null Distribution
.................................137
5.2.4
The Watson Test
.................................142
5.3
Generalisations of EDF Tests
............................144
5.3.1
Tests Based on the Empirical Quantile Function
(EQF)
..........................................145
5.3.2
Tests Based on the Empirical Characteristic Function
(ECF)
..........................................151
5.3.3
Miscellaneous Tests Based on Empirical Functionals
of
F
............................................153
5.4
The Sample Space Partition Tests
........................155
5.4.1
Another Look at the Anderson-Darling Statistic
.....155
5.4.2
The Sample Space Partition Test
...................155
5.5
Some Further Bibliographic Notes
........................158
5.6
Some Practical Guidelines for EDF Tests
..................159
Part II Two-Sample and If-Sample Problems
6
Introduction
..............................................163
6.1
The Problem Defined
....................................164
6.1.1
The Null Hypothesis of the General Two-Sample
Problem
.........................................164
6.1.2
The Null Hypothesis of the General ff-Sample
Problem
.........................................165
6.2
Example
Datasets
......................................166
6.2.1
Gene Expression in Colorectal Cancer Patients
.......166
6.2.2
Travel Times
....................................lß7
Contents
Preliminaries
(Building Blocks)...........................171
7.1 Permutation Tests......................................171
7.1.1
Introduction by Example
..........................171
7.1.2
Some Permutation and Randomisation Test Theory.
.. 175
7.2
Linear Rank Tests
......................................179
7.2.1
Simple Linear Rank Statistics
......................179
7.2.2
Locally Most Powerful Linear Rank Tests
...........187
7.2.3
Adaptive Linear Rank Tests
.......................190
7.3
The Pooled Empirical Distribution Function
...............190
7.4
The Comparison Distribution
............................191
7.5
The Quantile Process
...................................192
7.5.1
Contrast Processes
...............................192
7.5.2
Comparison Distribution Processes
.................194
7.6
Stochastic Ordering and Related Properties
................196
Graphical Tools
...........................................201
8.1
PP and QQ Plots
......................................201
8.1.1
Population Plots
.................................201
8.1.2
Empirical PP and QQ Plots
.......................205
8.2
Comparisons Distributions
...............................213
8.2.1
The Population Comparison Distribution
............213
8.2.2
The Empirical Comparison Distribution
.............213
Some Important Two-Sample Tests
.......................221
9.1
The Relation Between Statistical Tests and Hypotheses
.....222
9.1.1
Introduction
.....................................222
9.2
The Wilcoxon Rank Sum and the Mann-Whitney Tests
.....225
9.2.1
Introduction
.....................................225
9.2.2
The Hypotheses
..................................226
9.2.3
The Test Statistics
...............................227
9.2.4
The Null Distribution
.............................228
9.2.5
The WMW Test as a LMPRT
.....................230
9.2.6
The MW Statistic as an Estimator of
π
.............232
9.2.7
The Hodges-Lehmann Estimator
...................234
9.2.8
Examples
.......................................234
9.3
The Diagnostic Property of Two-Sample Tests
.............243
9.3.1
The Semiparametric Framework
....................244
9.3.2
Natural and Implied Null Hypotheses
...............246
9.3.3
The WMW Test in the Semiparametric Framework
... 246
9.3.4
Empirical Variance Estimators of Simple Linear Rank
Statistics
........................................250
9.4
Optimal Linear Rank Tests for Normal Location-Shift
Models
................................................253
9.5
Rank Tests for Scale Differences
..........................254
Contents
xvii
9.5.1
The Scale-Difference Model
........................255
9.5.2
The Capon and
Klotz
Tests
........................256
9.5.3
Some Other Important Tests
.......................257
9.5.4
Conclusion
......................................265
9.6
The Kruskal-Wallis Test and the ANOVA
F
-Test
..........265
9.6.1
The Hypotheses and the Test Statistic
..............266
9.6.2
The Null Distribution
.............................267
9.6.3
The Diagnostic Property
..........................267
9.6.4
The F-Test in ANOVA
............................268
9.7
Some Final Remarks
....................................269
9.7.1
Adaptive Tests
...................................269
9.7.2
The Lepage Test
.................................270
10
Smooth Tests
.............................................271
10.1
Smooth Tests for the 2-Sample Problem
...................271
10.1.1
Smooth Models and the Smooth Test
...............271
10.1.2
Components
.....................................275
10.2
The Diagnostic Property
................................278
10.2.1
Examples
.......................................279
10.3
Smooth Tests for the if-SampIe Problem
..................282
10.3.1
Smooth Models and the Smooth Test
...............282
10.3.2
Components
.....................................286
10.4
Adaptive Smooth Tests
.................................288
10.4.1
Order Selection and Subset Selection with a Finite
Horizon
.........................................288
10.4.2
Order Selection with an Infinite Horizon
.............289
10.5
Examples
..............................................290
10.6
Smooth Tests That Are Not Based on Ranks
...............294
10.7
Some Practical Guidelines for Smooth Tests
...............295
11
Methods Based on the Empirical Distribution Function
... 297
11.1
The Two-Sample and if-Sample Kolmogorov-Smirnov Test
.. 297
11.1.1
The Kolmogorov-Smirnov Test for the Two-Sample
Problem
.........................................297
11.1.2
The Kolmogorov-Smirnov Test for the
ÜT-Sample
Problem
.........................................299
11.2
Tests of the Anderson-Darling Type
......................299
11.2.1
The Test Statistic
................................299
11.2.2
The Components
.................................301
11.2.3
The Null Distribution
.............................303
11.2.4
Examples
.......................................304
xviii
Contents
11.3
Adaptive Tests of Neuhaus
..............................306
11.3.1
The General Idea
.................................306
11.3.2
Smooth Tests
....................................308
11.3.3
EDF tests
.......................................308
11.4
Some Practical Guidelines for EDF Tests
..................309
12
Two Final Methods and Some Final Thoughts
............311
12.1
A Contigency Table Approach
...........................311
12.2
The Sample Space Partition Tests
........................313
12.3
Some Final Thoughts and Conclusions
....................315
A Proofs
....................................................321
A.I Proof of Theorem
1.1...................................321
A.2 Proof of Theorem
1.2...................................322
A.3 Proof of Theorem
4.1 ...................................323
A.4 Proof of Lemma
4.1.....................................324
A.5 Proof of Lemma
4.2.....................................325
A.6 Proof of Lemma
4.3.....................................325
A.7 Proof of Theorem
4.10..................................326
A.8 Proof of Theorem
4.2...................................326
A.9 Heuristic Proof of Theorem
5.2...........................331
A.10 Proof of Theorem
9.1...................................332
В
The Bootstrap and Other Simulation Techniques
.........335
B.I Simulation of EDF Statistics Under the Simple Null
Hypothesis
............................................335
B.2 The Parametric Bootstrap for Composite Null Hypotheses
... 336
B.3 A Modified Nonparametric Bootstrap for Testing
Semiparametric Null Hypotheses
.........................336
References
....................................................339
Index
.........................................................351
|
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| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.24 |
| dewey-search | 519.24 |
| dewey-sort | 3519.24 |
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| id | DE-604.BV025599615 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:37:15Z |
| institution | BVB |
| isbn | 9780387927091 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020195190 |
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| owner_facet | DE-11 DE-824 DE-384 DE-188 DE-521 DE-355 DE-BY-UBR DE-703 |
| physical | XVIII, 353 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spelling | Thas, Olivier Verfasser aut Comparing distributions Olivier Thas New York, NY Springer 2010 XVIII, 353 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Literaturverz. S. 339 - 349 Statistik (DE-588)4056995-0 gnd rswk-swf Anpassungstest (DE-588)4208021-6 gnd rswk-swf Statistik (DE-588)4056995-0 s Anpassungstest (DE-588)4208021-6 s DE-604 Erscheint auch als Online-Ausgabe 978-0-387-92710-7 V:DE-604 application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=018011205&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020195190&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Thas, Olivier Comparing distributions Statistik (DE-588)4056995-0 gnd Anpassungstest (DE-588)4208021-6 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4208021-6 |
| title | Comparing distributions |
| title_auth | Comparing distributions |
| title_exact_search | Comparing distributions |
| title_full | Comparing distributions Olivier Thas |
| title_fullStr | Comparing distributions Olivier Thas |
| title_full_unstemmed | Comparing distributions Olivier Thas |
| title_short | Comparing distributions |
| title_sort | comparing distributions |
| topic | Statistik (DE-588)4056995-0 gnd Anpassungstest (DE-588)4208021-6 gnd |
| topic_facet | Statistik Anpassungstest |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=018011205&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020195190&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT thasolivier comparingdistributions |
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