A course in mathematical statistics and large sample theory:
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| Main Authors: | , , |
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| Format: | Electronic eBook |
| Language: | English |
| Published: |
New York
Springer
[2016]
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| Series: | Springer texts in statistics
|
| Subjects: | |
| Online Access: | BTU01 FHR01 FRO01 TUM01 UBM01 UBT01 UBW01 UEI01 UPA01 Volltext Inhaltsverzeichnis |
| Physical Description: | 1 Online-Ressource (XI, 389 Seiten, 9 illus., 2 illus. in color) |
| ISBN: | 9781493940325 |
| ISSN: | 1431-875X |
| DOI: | 10.1007/978-1-4939-4032-5 |
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| _version_ | 1804176551780024320 |
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| adam_text | Titel: A course in mathematical statistics and large sample theory
Autor: Bhattacharya, Rabindra N
Jahr: 2016
Contents
Part I Mathematical Statistics: Basic (Nonasymptotic) Theory
1 Introduction......................................................................................................3
1.1 What is Statistical Inference?..................................................................3
1.2 Sampling Schemes......................................................................................4
1.3 Some Simple Examples of Inference........................................................6
1.4 Notes and References................................................................................7
Exercises ..............................................................................................................8
Reference..............................................................................................................9
2 Decision Theory..............................................................................................11
2.1 Decision Rules and Risk Functions..........................................................11
2.2 Randomized Decision Rules, Admissibility............................................15
2.3 Notes and References................................................................................16
Exercises ..............................................................................................................16
References............................................................................................................17
3 Introduction to General Methods of Estimation..............................19
3.1 The Maximum Likelihood Estimator......................................................19
3.2 Method of Moments..................................................................................21
3.3 Bayes Rules and Bayes Estimators..........................................................22
3.4 Minimax Decision Rules............................................................................30
3.5 Generalized Bayes Rules and the James-Stein Estimator....................32
3.6 Notes and References................................................................................35
Exercises ..............................................................................................................35
References............................................................................................................37
4 Sufficient Statistics, Exponential Families, and Estimation..........39
4.1 Sufficient Statistics and Unbiased Estimation ......................................39
4.2 Exponential Families..................................................................................47
4.3 The Cramer-Rao Inequality ....................................................................55
4.4 Notes and References................................................................................59
Exercises ..............................................................................................................60
vii
Contents
viii
62
A Project for Students.......................... • -----; Q
Appendix for Project: The Nonparametric Percentile Bootstrap of Efron 6^
References.................................................
5 Testing Hypotheses.......................................
5.1 Introduction...........................................
5.2 Simple Hypotheses and the Neyman-Pearson Lemma.......
5.3 Examples..............................................
5.4 The Generalized N-P Lemma and UMP Unbiased Tests.....
5.5 UMP Unbiased Tests in the Presence of Nuisance Parameters
5.5.1 UMPU Tests in fc-Parameter Exponential Families----
5.6 Basu s Theorem........................................
5.7 Duality Between Tests and Confidence Regions.............
5.8 Invariant Tests, the Two-Sample Problem and Rank Tests .. .
5.8.1 The Two-Sample Problem.........................
5.9 Linear Models..........................................
5.9.1 The Gauss-Markov Theorem.......................
5.9.2 Testing in Linear Models..........................
5.10 Notes and References...................................
Exercises ..................................................
References.................................................
Part II Mathematical Statistics: Large Sample Theory
6 Consistency and Asymptotic Distributions of Statistics........117
6.1 Introduction................................................117
6.2 Almost Sure Convergence, Convergence in Probability
and Consistency of Estimators................................117
6.3 Consistency of Sample Moments and Regression Coefficients......119
6.4 Consistency of Sample Quantiles..............................126
6.5 Convergence in Distribution or in Law (or Weak Convergence):
The Central Limit Theorem ..................................128
6.6 Asymptotics of Linear Regression..............................134
6.7 Asymptotic Distribution of Sample Quantiles, Order Statistics____143
6.8 Asymptotics of Semiparametric Multiple Regression.............146
6.9 Asymptotic Relative Efficiency (ARE) of Estimators.............151
6.10 Constructing (Nonparametric) Confidence Intervals..............153
6.11 Errors in Variables Models....................................154
6.12 Notes and References........................................157
Exercises ..............................................
References..........................................1 ^ 4
7 Large Sample Theory of Estimation in Parametric Models
7.1 Introduction...................................
7.2 The Cramer-Rao Bound..............................
7.3 Maximum Likelihood: The One Parameter Case............
7.4 The Multi-Parameter Case..............................
7.5 Method of Moments..............................
7.6 Asymptotic Efficiency of Bayes Estimators.................
7.7 Asymptotic Normality of Af-estimators...................
67
67
69
70
74
79
81
84
88
92
94
97
97
99
109
110
112
165
165
166
168
174
185
189
191
Contents ix
7.8 Asymptotic Efficiency and Super Efficiency.....................194
Exercises .......................................................196
References......................................................200
8 Tests in Parametric and Nonparametric Models...............203
8.1 Pitman ARE (Asymptotic Relative Efficiency) ..................203
8.2 CLT for {/-Statistics and Some Two-Sample Rank Tests..........208
8.3 Asymptotic Distribution Theory of Parametric Large Sample Tests 215
8.4 Tests for Goodness-of-Fit.....................................222
8.5 Nonparametric Inference for the Two-Sample Problem...........228
8.6 Large Sample Theory for Stochastic Processes...................233
8.7 Notes and References........................................250
Exercises .......................................................251
References......................................................255
9 The Nonparametric Bootstrap.................................257
9.1 What is Bootstrap ? Why Use it?............................257
9.2 When Does Bootstrap Work?.................................259
9.2.1 Linear Statistics, or Sample Means......................259
9.2.2 Smooth Functions of Sample Averages...................260
9.2.3 Linear Regression.....................................261
9.3 Notes and References........................................264
Exercises .......................................................264
References......................................................265
10 Nonparametric Curve Estimation..............................267
10.1 Nonparametric Density Estimation............................267
10.2 Nonparametric Regression-Kernel Estimation...................272
10.3 Notes and References........................................276
Exercises .......................................................276
References......................................................277
Part III Special Topics
11 Edgeworth Expansions and the Bootstrap .....................281
11.1 Cramer Type Expansion for the Multivariate CLT...............281
11.2 The Formal Edgeworth Expansion and Its Validity ..............282
11.3 Bootstrap and Edgeworth Expansion...........................289
11.4 Miscellaneous Applications ...................................293
11.4.1 Cornish-Fisher Expansions .............................293
11.4.2 Higher Order Efficiency................................294
11.4.3 Computation of Power in Parametric Models..............294
11.4.4 Convergence of Markov Processes to Diffusions............294
11.4.5 Asymptotic Expansions in Analytic Number Theory.......294
11.4.6 Asymptotic Expansions for Time Series..................295
11.5 Notes and References........................................295
Exercises .......................................................299
References......................................................299
Contents
x
12 FnSchet Means and Nonparametric Inference on Non
Euclidean Geometric Spaces.........................
12.1 Introduction.....................................
12.2 Frechet Means on Metric Spaces.....................
12.3 Data Examples...................................
12.4 Notes and References..............................
Exercises............................................
References...........................................
13 Multiple Testing and the False Discovery Rate......
13.1 Introduction......................................
13.2 False Discovery Rate...............................
13.3 An Application to a Diffusion Tensor Imaging Data Set
13.4 Notes and References..............................
Exercises ............................................
References...........................................
14 Markov Chain Monte Carlo (MCMC) Simulation and Bayes
Theory........................................................325
14.1 Metropolis-Hastings Algorithm................................325
14.2 Gibbs Sampler..............................................327
14.3 Bayes Estimation in the Challenger Disaster Problem: A Project
for Students................................................329
A Project for Students............................................330
14.4 Notes and References........................................331
Exercises .......................................................331
References......................................................331
15 Miscellaneous Topics ..........................................333
15.1 Classification/Machine Learning...............................333
15.2 Principal Component Analysis (PCA)..........................335
15.3 Sequential Probability Ratio Test (SPRT)......................337
15.4 Notes and References........................................340
Exercises .......................................................340
References................................................................941
Appendices..................................
Standard Distributions..................
A.l Standard Univariate Discrete Distributions
A.2 Some Absolutely Continuous Distributions
A.2.1 The Normal Distribution N(/i, cr2).
A.3 The Multivariate Normal Distribution____
Exercises .................................
Moment Generating Functions (M.G.F.).....
303
303
304
311
313
313
314
317
317
318
321
321
322
322
343
343
345
347
352
356
Contents xi
Computation of Power of Some Optimal Tests: Non-central £, y2
and F .........................................................363
Liapounov s, Lindeberg s and Polya s Theorems ...................369
Solutions of Selected Exercises in Part I...........................371
Index..............................................................385
|
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| author | Bhattacharya, Rabindra N. 1937- Lin, Lizhen Patrângenaru, Victor |
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| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4939-4032-5 |
| format | Electronic eBook |
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| id | DE-604.BV043746282 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:33:59Z |
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| isbn | 9781493940325 |
| issn | 1431-875X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029157920 |
| oclc_num | 958035758 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-20 DE-739 DE-634 DE-898 DE-BY-UBR DE-861 DE-703 DE-824 DE-83 |
| owner_facet | DE-91 DE-BY-TUM DE-19 DE-BY-UBM DE-20 DE-739 DE-634 DE-898 DE-BY-UBR DE-861 DE-703 DE-824 DE-83 |
| physical | 1 Online-Ressource (XI, 389 Seiten, 9 illus., 2 illus. in color) |
| psigel | ZDB-2-SMA ZDB-2-SMA_2016 |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer texts in statistics |
| spelling | Bhattacharya, Rabindra N. 1937- Verfasser (DE-588)120600188 aut A course in mathematical statistics and large sample theory Rabi Bhattacharya, Lizhen Lin, Victor Patrangenaru New York Springer [2016] © 2016 1 Online-Ressource (XI, 389 Seiten, 9 illus., 2 illus. in color) txt rdacontent c rdamedia cr rdacarrier Springer texts in statistics 1431-875X Statistics Mathematical statistics Biostatistics Probabilities Statistical Theory and Methods Probability and Statistics in Computer Science Statistics for Business/Economics/Mathematical Finance/Insurance Probability Theory and Stochastic Processes Statistics and Computing/Statistics Programs Statistik Statistik (DE-588)4056995-0 gnd rswk-swf Statistik (DE-588)4056995-0 s 1\p DE-604 Lin, Lizhen Verfasser (DE-588)159545404 aut Patrângenaru, Victor Verfasser aut Erscheint auch als Druckausgabe 978-1-4939-4030-1 https://doi.org/10.1007/978-1-4939-4032-5 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029157920&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bhattacharya, Rabindra N. 1937- Lin, Lizhen Patrângenaru, Victor A course in mathematical statistics and large sample theory Statistics Mathematical statistics Biostatistics Probabilities Statistical Theory and Methods Probability and Statistics in Computer Science Statistics for Business/Economics/Mathematical Finance/Insurance Probability Theory and Stochastic Processes Statistics and Computing/Statistics Programs Statistik Statistik (DE-588)4056995-0 gnd |
| subject_GND | (DE-588)4056995-0 |
| title | A course in mathematical statistics and large sample theory |
| title_auth | A course in mathematical statistics and large sample theory |
| title_exact_search | A course in mathematical statistics and large sample theory |
| title_full | A course in mathematical statistics and large sample theory Rabi Bhattacharya, Lizhen Lin, Victor Patrangenaru |
| title_fullStr | A course in mathematical statistics and large sample theory Rabi Bhattacharya, Lizhen Lin, Victor Patrangenaru |
| title_full_unstemmed | A course in mathematical statistics and large sample theory Rabi Bhattacharya, Lizhen Lin, Victor Patrangenaru |
| title_short | A course in mathematical statistics and large sample theory |
| title_sort | a course in mathematical statistics and large sample theory |
| topic | Statistics Mathematical statistics Biostatistics Probabilities Statistical Theory and Methods Probability and Statistics in Computer Science Statistics for Business/Economics/Mathematical Finance/Insurance Probability Theory and Stochastic Processes Statistics and Computing/Statistics Programs Statistik Statistik (DE-588)4056995-0 gnd |
| topic_facet | Statistics Mathematical statistics Biostatistics Probabilities Statistical Theory and Methods Probability and Statistics in Computer Science Statistics for Business/Economics/Mathematical Finance/Insurance Probability Theory and Stochastic Processes Statistics and Computing/Statistics Programs Statistik |
| url | https://doi.org/10.1007/978-1-4939-4032-5 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029157920&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bhattacharyarabindran acourseinmathematicalstatisticsandlargesampletheory AT linlizhen acourseinmathematicalstatisticsandlargesampletheory AT patrangenaruvictor acourseinmathematicalstatisticsandlargesampletheory |