Theory of statistics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1997
|
| Edition: | Corr. 2. print. |
| Series: | Springer series in statistics
|
| Subjects: | |
| Links: | http://www.loc.gov/catdir/enhancements/fy0815/97115378-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017159713&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke Includes bibliographical references (p. [675]-688) and indexes |
| Physical Description: | XVI, 702 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9781461287087 0387945466 |
Staff View
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| 245 | 1 | 0 | |a Theory of statistics |c Mark J. Schervish |
| 250 | |a Corr. 2. print. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1997 | |
| 300 | |a XVI, 702 S. |b Ill., graph. Darst. |c 24 cm | ||
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Record in the Search Index
| _version_ | 1819334600793522176 |
|---|---|
| adam_text | Contents
Preface vii
Chapter 1: Probability Models 1
1.1 Background 1
1.1.1 General Concepts 1
1.1.2 Classical Statistics 2
1.1.3 Bayesian Statistics 4
1.2 Exchangeability 5
1.2.1 Distributional Symmetry 5
1.2.2 Frequency and Exchangeability 10
1.3 Parametric Models 12
1.3.1 Prior, Posterior, and Predictive Distributions 13
1.3.2 Improper Prior Distributions 19
1.3.3 Choosing Probability Distributions 21
1.4 DeFinetti s Representation Theorem 24
1.4.1 Understanding the Theorems 24
1.4.2 The Mathematical Statements 26
1.4.3 Some Examples 28
1.5 Proofs of DeFinetti s Theorem and Related Results* 33
1.5.1 Strong Law of Large Numbers 33
1.5.2 The Bernoulli Case 36
1.5.3 The General Finite Case* 38
1.5.4 The General Infinite Case 45
1.5.5 Formal Introduction to Parametric Models* 49
1.6 Infinite-Dimensional Parameters* 52
1.6.1 Dirichlet Processes 52
1.6.2 Tailfree Processes 1 60
1.7 Problems 73
Chapter 2: Sufficient Statistics 82
2.1 Definitions 82
2.1.1 Notational Overview 82
2.1.2 Sufficiency 83
2.1.3 Minimal and Complete Sufficiency 92
2.1.4 Ancillarity 95
2.2 Exponential Families of Distributions 102
* Sections and chapters marked with an asterisk may be skipped or covered
out of order without interrupting the flow of ideas.
+Sections marked with a plus sign include results which rely on the theory of
martingales. They may be skipped without interrupting the flow of ideas.
xii Contents
2.2.1 Basic Properties 102
2.2.2 Smoothness Properties 105
2.2.3 A Characterization Theorem* 109
2.3 Information HO
2.3.1 Fisher Information Ill
2.3.2 Kullback-Leibler Information 115
2.3.3 Conditional Information* 118
2.3.4 Jeffreys Prior* 121
2.4 Extremal Families* 123
2.4.1 The Main Results 124
2.4.2 Examples 127
2.4.3 Proofs 1 129
2.5 Problems 138
Chapter 3: Decision Theory 144
3.1 Decision Problems 144
3.1.1 Framework 144
3.1.2 Elements of Bayesian Decision Theory 146
3.1.3 Elements of Classical Decision Theory 149
3.1.4 Summary 150
3.2 Classical Decision Theory 150
3.2.1 The Role of Sufficient Statistics 150
3.2.2 Admissibility 153
3.2.3 James-Stein Estimators 163
3.2.4 Minimax Rules 167
3.2.5 Complete Classes 174
3.3 Axiomatic Derivation of Decision Theory* 181
3.3.1 Definitions and Axioms 181
3.3.2 Examples 186
3.3.3 The Main Theorems 188
3.3.4 Relation to Decision Theory 189
3.3.5 Proofs of the Main Theorems* 190
3.3.6 State-Dependent Utility* 205
3.4 Problems 208
Chapter 4: Hypothesis Testing 214
4.1 Introduction 214
4.1.1 A Special Kind of Decision Problem 214
4.1.2 Pure Significance Tests 216
4.2 Bayesian Solutions 218
4.2.1 Testing in General 218
4.2.2 Bayes Factors 220
4.3 Most Powerful Tests 230
4.3.1 Simple Hypotheses and Alternatives 233
4.3.2 Simple Hypotheses, Composite Alternatives 238
4.3.3 One-Sided Tests 239
4.3.4 Two-Sided Hypotheses 246
4.4 Unbiased Tests 253
4.4.1 General Results 253
Contents xiii
4.4.2 Interval Hypotheses 255
4.4.3 Point Hypotheses 257
4.5 Nuisance Parameters 265
4.5.1 Neyman Structure 265
4.5.2 Tests about Natural Parameters 268
4.5.3 Linear Combinations of Natural Parameters 272
4.5.4 Other Two-Sided Cases* 272
4.5.5 Likelihood Ratio Tests 274
4.5.6 The Standard F-Test as a Bayes Rule* 276
4.6 P- Values 279
4.6.1 Definitions and Examples 279
4.6.2 P-Values and Bayes Factors 283
4.7 Problems 285
Chapter 5: Estimation 296
5.1 Point Estimation 296
5.1.1 Minimum Variance Unbiased Estimation 297
5.1.2 Lower Bounds on the Variance of Unbiased Estimators . . . 301
5.1.3 Maximum Likelihood Estimation 307
5.1.4 Bayesian Estimation 309
5.1.5 Robust Estimation* 310
5.2 Set Estimation 315
5.2.1 Confidence Sets 315
5.2.2 Prediction Sets* 324
5.2.3 Tolerance Sets* 325
5.2.4 Bayesian Set Estimation 327
5.2.5 Decision Theoretic Set Estimation* 328
5.3 The Bootstrap* 329
5.3.1 The General Concept 329
5.3.2 Standard Deviations and Bias 335
5.3.3 Bootstrap Confidence Intervals 336
5.4 Problems 339
Chapter 6: Equivariance* 344
6.1 Common Examples 344
6.1.1 Location Problems 344
6.1.2 Scale Problems* 350
6.2 Equivariant Decision Theory 353
6.2.1 Groups of Transformations 353
6.2.2 Equivariance and Changes of Units 359
6.2.3 Minimum Risk Equivariant Decisions 363
6.3 Testing and Confidence Intervals* 375
6.3.1 P-Values in Invariant Problems 375
6.3.2 Equivariant Confidence Sets 379
6.3.3 Invariant Tests* 380
6.4 Problems 388
xiv Contents
Chapter 7: Large Sample Theory 394
7.1 Convergence Concepts 394
7.1.1 Deterministic Convergence 394
7.1.2 Stochastic Convergence 395
7.1.3 The Delta Method 401
7.2 Sample Quantiles 404
7.2.1 A Single Quantile 404
7.2.2 Several Quantiles 408
7.2.3 Linear Combinations of Quantiles* 410
7.3 Large Sample Estimation 412
7.3.1 Some Principles of Large Sample Estimation 412
7.3.2 Maximum Likelihood Estimators 415
7.3.3 MLEs in Exponential Families 418
7.3.4 Examples of Inconsistent MLEs 420
7.3.5 Asymptotic Normality of MLEs 421
7.3.6 Asymptotic Properties of M-Estimators* 424
7.4 Large Sample Properties of Posterior Distributions 428
7.4.1 Consistency of Posterior Distributions 1^ 429
7.4.2 Asymptotic Normality of Posterior Distributions 435
7.4.3 Laplace Approximations to Posterior Distributions* .... 446
7.4.4 Asymptotic Agreement of Predictive Distributions 1 .... 455
7.5 Large Sample Tests 458
7.5.1 Likelihood Ratio Tests 458
7.5.2 Chi-Squared Goodness of Fit Tests 461
7.6 Problems 467
Chapter 8: Hierarchical Models 476
8.1 Introduction 476
8.1.1 General Hierarchical Models 476
8.1.2 Partial Exchangeability* 479
8.1.3 Examples of the Representation Theorem* 480
8.2 Normal Linear Models 483
8.2.1 One-Way ANOVA 483
8.2.2 Two-Way Mixed Model ANOVA* 488
8.2.3 Hypothesis Testing 491
8.3 Nonnormal Models* 495
8.3.1 Poisson Process Data 495
8.3.2 Bernoulli Process Data 497
8.4 Empirical Bayes Analysis* 500
8.4.1 Naive Empirical Bayes 500
8.4.2 Adjusted Empirical Bayes 503
8.4.3 Unequal Variance Case 504
8.5 Successive Substitution Sampling 505
8.5.1 The General Algorithm 505
8.5.2 Normal Hierarchical Models 512
8.5.3 Nonnormal Models 517
8.6 Mixtures of Models 519
8.6.1 General Mixture Models 519
8.6.2 Outliers 521
Contents xv
8.6.3 Bayesian Robustness 524
8.7 Problems 532
Chapter 9: Sequential Analysis 536
9.1 Sequential Decision Problems 536
9.2 The Sequential Probability Ratio Test 548
9.3 Interval Estimation* 558
9.4 The Relevance of Stopping Rules 562
9.5 Problems 567
Appendix A: Measure and Integration Theory 570
A.I Overview 570
A.I.I Definitions 570
A.1.2 Measurable Functions 572
A.I.3 Integration 573
A.1.4 Absolute Continuity 574
A.2 Measures 575
A.3 Measurable Functions 582
A.4 Integration 587
A.5 Product Spaces 593
A.6 Absolute Continuity 597
A.7 Problems 602
Appendix B: Probability Theory 606
B.I Overview 606
B.I.I Mathematical Probability 606
B.1.2 Conditioning 607
B.1.3 Limit Theorems 611
B.2 Mathematical Probability 612
B.2.1 Random Quantities and Distributions 612
B.2.2 Some Useful Inequalities 613
B.3 Conditioning 615
B.3.1 Conditional Expectations 615
B.3.2 Borel Spaces* 619
B.3.3 Conditional Densities 623
B.3.4 Conditional Independence 628
B.3.5 The Law of Total Probability 632
B.4 Limit Theorems 634
B.4.1 Convergence in Distribution and in Probability 634
B.4.2 Characteristic Functions 639
B.5 Stochastic Processes 645
B.5.1 Introduction 645
B.5.2 Martingales+ 645
B.5.3 Markov Chains* 650
B.5.4 General Stochastic Processes 651
B.6 Subjective Probability 654
B.7 Simulation* 659
B.8 Problems 661
xvi Contents
Appendix C: Mathematical Theorems Not Proven Here 665
C.I Real Analysis 665
C.2 Complex Analysis 666
C.3 Functional Analysis 667
Appendix D: Summary of Distributions 668
D.I Univariate Continuous Distributions 668
D.2 Univariate Discrete Distributions 672
D.3 Multivariate Distributions 674
References 675
Notation and Abbreviation Index 689
Name Index 691
Subject Index 694
|
| any_adam_object | 1 |
| author | Schervish, Mark J. |
| author_facet | Schervish, Mark J. |
| author_role | aut |
| author_sort | Schervish, Mark J. |
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| building | Verbundindex |
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| callnumber-subject | QA - Mathematics |
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| ctrlnum | (OCoLC)36556776 (DE-599)BVBBV035355643 |
| dewey-full | 519.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5 |
| dewey-search | 519.5 |
| dewey-sort | 3519.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | Corr. 2. print. |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV035355643 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:29:14Z |
| institution | BVB |
| isbn | 9781461287087 0387945466 |
| language | English |
| lccn | 97115378 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017159713 |
| oclc_num | 36556776 |
| open_access_boolean | |
| owner | DE-898 DE-BY-UBR DE-11 DE-92 DE-355 DE-BY-UBR |
| owner_facet | DE-898 DE-BY-UBR DE-11 DE-92 DE-355 DE-BY-UBR |
| physical | XVI, 702 S. Ill., graph. Darst. 24 cm |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spellingShingle | Schervish, Mark J. Theory of statistics Inferencia estatistica larpcal Mathematical statistics Statistischer Test (DE-588)4077852-6 gnd Varianzanalyse (DE-588)4187413-4 gnd Statistik (DE-588)4056995-0 gnd Auswahlfunktion (DE-588)4134349-9 gnd Entscheidungstheorie (DE-588)4138606-1 gnd Schätztheorie (DE-588)4121608-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4077852-6 (DE-588)4187413-4 (DE-588)4056995-0 (DE-588)4134349-9 (DE-588)4138606-1 (DE-588)4121608-8 (DE-588)4064324-4 (DE-588)4123623-3 |
| title | Theory of statistics |
| title_auth | Theory of statistics |
| title_exact_search | Theory of statistics |
| title_full | Theory of statistics Mark J. Schervish |
| title_fullStr | Theory of statistics Mark J. Schervish |
| title_full_unstemmed | Theory of statistics Mark J. Schervish |
| title_short | Theory of statistics |
| title_sort | theory of statistics |
| topic | Inferencia estatistica larpcal Mathematical statistics Statistischer Test (DE-588)4077852-6 gnd Varianzanalyse (DE-588)4187413-4 gnd Statistik (DE-588)4056995-0 gnd Auswahlfunktion (DE-588)4134349-9 gnd Entscheidungstheorie (DE-588)4138606-1 gnd Schätztheorie (DE-588)4121608-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Inferencia estatistica Mathematical statistics Statistischer Test Varianzanalyse Statistik Auswahlfunktion Entscheidungstheorie Schätztheorie Wahrscheinlichkeitsrechnung Lehrbuch |
| url | http://www.loc.gov/catdir/enhancements/fy0815/97115378-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017159713&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT schervishmarkj theoryofstatistics |