Nonparametric estimation under shape constraints: estimators, algorithms, and asymptotics
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
2014
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| Series: | Cambridge series in statistical and probabilistic mathematics
38 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (pages 401-408) and index |
| Physical Description: | XI, 416 S. Ill. 27 cm |
| ISBN: | 9780521864015 0521864011 |
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MARC
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| 264 | 1 | |c 2014 | |
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| 650 | 4 | |a Multivariate analysis | |
| 650 | 4 | |a Mathematical statistics | |
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| adam_text | Titel: Nonparametric estimation under shape constraints
Autor: Groeneboom, Piet
Jahr: 2014
Contents
Preface and Acknowledgments page ix
1 Introduction 1
1.1 Is There a Warming-up of Lake Mendota? 1
1.2 Onset ofNonlethal Lung Tumor 3
1.3 The Transmission Potential of a Disease 8
1.4 The Bangkok Cohort Study 10
1.5 Inverse Problems, Censoring, Mixture Models and
Shape Constraints 11
1.6 Outline of the Book 15
Exercises 16
Bibliographie Remarks 17
2 Basic Estimation Problems with Monotonicity Constraints 18
2.1 Monotone Regression 18
2.2 Monotone Density Estimation 22
2.3 Estimating a Distribution Function ffom Current Status Data 28
2.4 Deconvolution Problem with Jump Kernel 31
2.5 Generalized Isotonic Regression Problems 33
2.6 Estimating a Monotone Hazard Rate 37
Exercises 42
Bibliographie Remarks 46
3 Asymptotic Theory for the Basic Monotone Problems 47
3.1 Consistency 47
3.2 Heuristic Asymptotics for the Grenander Estimator 51
3.3 Convex Minorants: Basic Properties 55
3.4 Some Empirical Process Theory 57
3.5 Asymptotic Distribution in Exponential Deconvolution Model 60
3.6 Limit Distribution of the Grenander Estimator 64
3.7 Some Martingale Theory 67
3.8 Asymptotic Distribution of the MLE in the Current
Status Model 68
v
vi
Contents
3.9 Chernoff s Distribution 73
3.10 The Concave Maj orant of Brownian Motion and
Brownian Bridge 77
Exercises 80
Bibliographie Remarks 86
4 Other Univariate Problems Involving
Monotonicity Constraints 87
4.1 Wicksell s Corpuscle Problem 87
4.2 Convex Regression 91
4.3 Convex Density Estimation 93
4.4 Log Concave Densities 101
4.5 Star Shaped Distributions on [0,1] 105
4.6 Deconvolution Problems 108
4.7 Interval Censoring Case 2 112
Exercises 116
Bibliographie Remarks 119
5 Higher Dimensional Problems 121
5.1 Competing Risks with Current Status Observations 121
5.2 Bivariate Interval Censoring 127
5.3 Current Status with Continuous Marks 132
5.4 Multivariate Log Concave Densities 134
Exercises 136
Bibliographie Remarks 138
6 Lower Bounds on Estimation Rates 139
6.1 Global and Local Minimax Risk 139
6.2 A Minimax Lower Bound Theorem 141
6.3 Lower Bound Based on the Van Trees Inequality 144
6.4 Applications 149
Exercises 152
Bibliographie Remarks 154
7 Algorithms and Computation 155
7.1 Algorithm: Concept and Convergence 156
7.2 The EM Algorithm 159
7.3 The Iterative Convex Minorant Algorithm 170
7.4 Vertex Direction Algorithms 177
7.5 The MLE in the Competing Risks Model with
Interval Censoring 186
Exercises 192
Bibliographie Remarks 195
Contents
vii
8 Shape and Smoothness 197
8.1 Smoothing a Shape-Constrained Estimator 198
8.2 Maximizing a Smoothed Objective Function 200
8.3 Penalized M-Estimation 204
8.4 Monotonie Rearrangements of Smooth Estimators 208
8.5 Maximum Smoothed Likelihood Estimation in the Current
Status Model 210
8.6 Smooth Estimation for Interval Censoring Case 2 215
8.7 Smooth Estimation in the Bivariate Interval Censoring Model 219
Exercises 222
Bibliographie Remarks 225
9 Testing and Confidence Intervals 226
9.1 Testing for a Monotone Hazard 227
9.2 Sample Tests for Decreasing Densities 238
9.3 Two-Sample Tests for Current Status Data 250
9.4 Two-Sample Tests for Case 2 Interval Censored Data 263
9.5 Pointwise Confidence Intervals for the Current Status Model 266
Exercises 279
Bibliographie Remarks 281
10 Asymptotic Theory of Smooth Functionals 283
10.1 Estimating the Expectation in Deconvolution Models 284
10.2 Estimating Smooth Functionals in the Current Status Model 286
10.3 The Integral Equation for Interval Censoring Case 2 291
10.4 Smooth Functional Estimation in the Interval Censoring
Case 2 Model 299
Exercises 309
Bibliographie Remarks 312
11 Pointwise Asymptotic Distribution Theory for
Univariate Problems 313
11.1 The LS Estimator of a Convex Density 313
11.2 Tail Bounds for the MLE in the Current Status Model 320
11.3 The SMLE in the Current Status Model 327
11.4 The SMLE for Interval Censoring Case 2 338
11.5 The MSLE for Interval Censoring Case 2 344
11.6 Estimation of a Nondecreasing Hazard in the Right Censoring
Model: SMLE 347
Exercises 356
Bibliographie Remarks 358
viii
Contents
12 Pointwise Asymptotic Distribution Theory for Multivariate
Problems 360
12.1 The ML Estimator in the Competing Risk Model
with Current Status Data 360
12.2 The SMLE in the Current Status Competing Risk Model 363
12.3 The Bivariate Current Status Model 369
Exercises 376
Bibliographie Remarks 377
13 Asymptotic Distribution of Global Deviations 378
13.1 The L Loss of the Grenander Estimator 378
13.2 Empirical L -Test for a Monotone Hazard 385
13.3 Two-Sample Tests for Current Status Data 392
Exercises 398
Bibliographie Remarks 399
References 401
Author Index 409
Subject Index 412
|
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| illustrated | Illustrated |
| indexdate | 2024-07-10T07:12:44Z |
| institution | BVB |
| isbn | 9780521864015 0521864011 |
| language | English |
| lccn | 014022148 |
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| oclc_num | 931610383 |
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| owner_facet | DE-11 |
| physical | XI, 416 S. Ill. 27 cm |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| record_format | marc |
| series | Cambridge series in statistical and probabilistic mathematics |
| series2 | Cambridge series in statistical and probabilistic mathematics |
| spelling | Groeneboom, Piet Verfasser aut Nonparametric estimation under shape constraints estimators, algorithms, and asymptotics Piet Groeneboom, Delft University of Technology, Geurt Jongbloed, Delft University of Technology 2014 XI, 416 S. Ill. 27 cm txt rdacontent n rdamedia nc rdacarrier Cambridge series in statistical and probabilistic mathematics 38 Includes bibliographical references (pages 401-408) and index Nonparametric statistics Multivariate analysis Mathematical statistics Estimation theory Jongbloed, Geurt 1968- Verfasser (DE-588)1065699875 aut Cambridge series in statistical and probabilistic mathematics 38 (DE-604)BV011442366 38 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028342894&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Groeneboom, Piet Jongbloed, Geurt 1968- Nonparametric estimation under shape constraints estimators, algorithms, and asymptotics Cambridge series in statistical and probabilistic mathematics Nonparametric statistics Multivariate analysis Mathematical statistics Estimation theory |
| title | Nonparametric estimation under shape constraints estimators, algorithms, and asymptotics |
| title_auth | Nonparametric estimation under shape constraints estimators, algorithms, and asymptotics |
| title_exact_search | Nonparametric estimation under shape constraints estimators, algorithms, and asymptotics |
| title_full | Nonparametric estimation under shape constraints estimators, algorithms, and asymptotics Piet Groeneboom, Delft University of Technology, Geurt Jongbloed, Delft University of Technology |
| title_fullStr | Nonparametric estimation under shape constraints estimators, algorithms, and asymptotics Piet Groeneboom, Delft University of Technology, Geurt Jongbloed, Delft University of Technology |
| title_full_unstemmed | Nonparametric estimation under shape constraints estimators, algorithms, and asymptotics Piet Groeneboom, Delft University of Technology, Geurt Jongbloed, Delft University of Technology |
| title_short | Nonparametric estimation under shape constraints |
| title_sort | nonparametric estimation under shape constraints estimators algorithms and asymptotics |
| title_sub | estimators, algorithms, and asymptotics |
| topic | Nonparametric statistics Multivariate analysis Mathematical statistics Estimation theory |
| topic_facet | Nonparametric statistics Multivariate analysis Mathematical statistics Estimation theory |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028342894&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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