简介
The Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists. "This is a nice book containing a wealth of information, much of it due to the authors. . . . If an instructor designing such a course wanted a textbook, this book would be the best choice available. . . . There are many stimulating exercises, and the book also contains an excellent index and an extensive list of references." - Technometrics "[This] book should be read carefully by anyone who is interested in dealing with statistical models in a realistic fashion." - American Scientist Introducing concepts, theory, and applications, Robust Statistics is accessible to a broad audience, avoiding allusions to high-powered mathematics while emphasizing ideas, heuristics, and background. The text covers the approach based on the influence function (the effect of an outlier on an estimater, for example) and related notions such as the breakdown point. It also treats the change-of-variance function, fundamental concepts and results in the framework of estimation of a single parameter, and applications to estimation of covariance matrices and regression parameters.
目录
Contents 17
1. INTRODUCTION AND MOTIVATION 27
1.1. The Place and Aims of Robust Statistics 27
1.1a. What Is Robust Statistics? 27
1.1b. The Relation to Some Other Key Words in Statistics 34
1.1c. The Aims of Robust Statistics 37
1.1d. An Example 40
1.2. Why Robust Statistics? 44
1.2a. The Role of Parametric Models 44
1.2b. Types of Deviations from Parametric Models 46
1.2c. The Frequency of Gross Errors 51
1.2d. The Effects of Mild Deviations from a Parametric Model 54
1.2e. How Necessary Are Robust Procedures? 57
1.3. The Main Approaches towards a Theory of Robustness 60
1.3a. Some Historical Notes 60
1.3b. Huber\u2019s Minimax Approach for Robust Estimation 62
1.3c. Huber\u2019s Second Approach to Robust Statistics via Robustifed Likelihood Ratio Tests 65
1.3d. The Approach Based on In Juence Functions 66
1.3e. The Relation between the Minimax Approach and the Approach Based on Influence Functions 73
1.3f. The Approach Based on Influence Functions as a Robustifed Likelihood Approach, and Its Relation to Various Statistical Schools 78
*1.4. Rejection of Outliers and Robust Statistics 82
1.4a. Why Rejection of Outliers? 82
1.4b. How Well Are Objective and Subjective Methods for the Rejection of Outliers Doing in the Context of Robust Estimation? 88
Exercises and Problems 97
2. ONE-DIMENSIONAL ESTIMATORS 104
2.0. An Introductory Example 104
2.1. The Influence Function 107
2.1a. Parametric Models, Estimators, and Functionals 107
2.1b. Definition and Properties of the Influence Function 109
2.1c. Robustness Measures Derived from the Influence Function 113
2.1d. Some Simple Examples 114
2.1e. Finite-Sample Versions 118
2.2. The Breakdown Point and Qualitative Robustness 122
2.2a. Global Reliability: The Breakdown Point 122
2.2b. Continuity and Qualitative Robustness 124
2.3. Classes of Estimators 126
2.3a. M-Estimators 126
2.3b. L-Estimators 134
2.3c. R-Estimators 136
2.3d. Other Types of Estimators: A, D, P, S, W 139
2.4. Optimally Bounding the Gross-Error Sensitivity 142
2.4a. The General Optimality Result 142
2.4b. M-Estimator 145
2.4c. L-Estimators 148
2.4d. R-Estimators 150
2.5. The Change-of-Variance Function 151
2.5a. Definitions 151
2.5b. B-Robustness versus V-Robustness 157
2.5c. The Most Robust Estimator 159
2.5d. Optimal Robust Estimators 160
2.5e. M-Estimators for Scale 165
*2.5f. Further Topics 170
2.6. Redescending M-Estimators 175
2.6a. Introduction 175
2.6b. Most Robust Estimators 180
2.6c. Optimal Robust Estimators 184
2.6d. Schematic Summary of Sections 2.5 and 2.6 194
*2.6e. Redescending M-Estimators for Scale 194
2.7. Relation with Huber\u2019s Minimax Approach 198
Exercises and Problems 204
3. ONE-DIMENSIONAL TESTS 213
3.1. Introduction 213
3.2. The Influence Function for Tests 215
3.2a. Definition of the Influence Function 215
3.2b. Properties of the Influence Function 220
3.2c. Relation with Level and Power 224
3.2d. Connection with Shift Estimators 228
3.3. Classes of Tests 230
3.3a The One-Sample Case 230
3.3b. The Two-Sample Case 232
3.4. Optimally Bounding the Gross-Error Sensitivity 235
3.5. Extending the Change-of-Variance Function to Tests 238
*3.6. Related Approaches 241
3.6a. Lambert\u2019s Approach 241
3.6b. Eplett\u2019s Approach 244
*3.7. M-Tests for a Simple Alternative 245
Exercises and Problems 247
4. MULTIDIMENSIONAL ESTIMATORS 251
4.1. Introduction 251
4.2. Concepts 252
4.2a. Influence Function 252
4.2b. Gross-Error Sensitivities 254
4.2c. M-Estimators 256
4.2d. Example: Location and Scale 258
4.3. Optimal Estimators 264
4.3a. The Unstandardized Case 264
4.3b. The Optimal B-Robust Estimators 269
4.3c. Existence and Uniqueness of the Optimal 蠄-Functions 272
4.3d How to Obtain Optimal Estimators 273
4.4. Partitioned Parameters 278
4.4a. Introduction: Location and Scale 278
*4.4b. Optimal Estimators 279
*4.5. Invariance 283
4.5a. Models Generated by Transformations 283
4.5b. Models and Invariance 283
4.5c. Equivariant Estimators 285
*4.6. Complements 286
4.6a. Admissible B-Robust Estimators 286
4.6b. Calculation of M-Estimates 289
Exercises and Problems 292
5. ESTIMATION OF COVARIANCE MATRICES AND MULTIVARIATE LOCATION 296
5.1. Introduction 296
5.2. The Model 297
5.2a. Definition 297
5.2b Scores 300
5.3. Equivariant Estimators 301
5.3a. Orthogonally Equivariant Vector Functions and d-Type Matrices 301
5.3b. General Results 306
5.3c. M-Estimators 309
5.4. Optimal and Most B-Robust Estimators 315
5.4a Full Parameter 315
5.4b. Partitioned Parameter 319
5.5. Breakdown Properties of Covariance Matrix Estimators 322
5.5a. Breakdown Point of M-Estimators 322
*5.5b. Breakdown at the Edge 325
5.5c. An Estimator with Breakdown Point 1/2 326
Exercises and Problems 329
6. LINEAR MODELS: ROBUST ESTIMATION 333
6.1. Introduction 333
6.1a. Overview 333
6.1b. The Model and the Classical Least-Squares Estimates 334
6.2. Huber-Estimators 337
6.3. M-Estimators for Linear Models 341
6.3a. Definition, Influence Function, and Sensitivities 341
6.3b. Most B-Robust and Optimal B-Robust Estimators 344
6.3c. The Change-of-Variance Function; Most V-Robust and Optimal V-Robust Estimators 349
6.4. Complements 354
6.4a. Breakdown Aspects 354
*6.4b. Asymptotic Behavior of Bounded Influence Estimators 357
6.4c. Computer Programs 361
*6.4d. Related Approaches 363
Exercises and Problems 364
7. LINEAR MODELS: ROBUST TESTING 368
7.1. Introduction 368
7.1a. Overview 368
7.lb. The Test Problem in Linear Models 369
7.2. A General Class of Tests for Linear Models 371
7.2a. Definition of 蟿Test 371
7.2b. Influence Function and Asymptotic Distribution 373
7.2c. Special Cases 380
7.3. Optimal Bounded Influence Tests 384
7.3a The pc-Test 384
7.3b The Optimal Mallows-Type Test 385
7.3c. The Optimal Test for the General M-Regression 386
7.3d. A Robust Procedure for Model Selection 392
*7.4. C(伪) \u2013Type Tests for Linear Models 393
7.4a. Definition of a C(伪) \u2013Type Test 394
7.4b. Influence Function and Asymptotic Power of C(伪)\u2013Type Tests 395
7.4c. Optimal Robust C(伪)\u2013Type Tests 399
7.4d Connection with an Asymptotically Minimax Test 399
7.5. Complements 402
*7.5a. Computation of Optimal 畏 Functions 402
*7.5b. Computation of the Asymptotic Distribution of the 蟿\u2013Test Statistic 403
*7.5c. Asymptotic Behavior of Direrent Tests for Simple Regression 404
7.5d. A Numerical Example 408
Exercises and Problems 411
8. COMPLEMENTS AND OUTLOOK 413
8.1. The Problem of Unsuspected Serial Correlations, or Violation of the Independence Assumption 413
8.la. Empirical Evidence for Semi-systematic Errors 413
8.1b. The Model of Self-similar Processes for Unsuspected Serial Correlations 415
8.1c. Some Consequences of the Model of Self-similar Processes 417
8.ld Estimation of the Long-Range Intensity of Serial Correlations 421
8.le. Some Further Problems of Robustness against Serial Correlations 422
8.2. Some Frequent Misunderstandings about Robust Statistics 423
8.2a. Some Common Objections against Huber\u2019s Minimax Approach 423
8.2b. \u201cRobust Statistics Is Not Necessary, Because ... \u201d 429
*8.2c. Some Details on Redescending Estimators 432
*8.2d. What Can Actually Be Estimated? 435
8.3. Robustness in Time Series 442
8.3a. Introduction 442
8.3b. The Influence Function for Time Series 443
8.3c. Other Robustness Problems in Time Series 448
*8.4. Some Special Topics Related to the Breakdown Point 448
8.4a. Most Robust (Median-Type) Estimators on the Real Line 448
8.4b. Special Structural Aspects of the Analysis of Variance 451
*8.5. Small-Sample Asymptotics 458
8.5a. Introduction 458
8.5b. Small-Sample Asymptotics for M-Estimators 459
8.5c. Further Applications 464
Exercises and Problems 464
REFERENCES 465
INDEX 491
1. INTRODUCTION AND MOTIVATION 27
1.1. The Place and Aims of Robust Statistics 27
1.1a. What Is Robust Statistics? 27
1.1b. The Relation to Some Other Key Words in Statistics 34
1.1c. The Aims of Robust Statistics 37
1.1d. An Example 40
1.2. Why Robust Statistics? 44
1.2a. The Role of Parametric Models 44
1.2b. Types of Deviations from Parametric Models 46
1.2c. The Frequency of Gross Errors 51
1.2d. The Effects of Mild Deviations from a Parametric Model 54
1.2e. How Necessary Are Robust Procedures? 57
1.3. The Main Approaches towards a Theory of Robustness 60
1.3a. Some Historical Notes 60
1.3b. Huber\u2019s Minimax Approach for Robust Estimation 62
1.3c. Huber\u2019s Second Approach to Robust Statistics via Robustifed Likelihood Ratio Tests 65
1.3d. The Approach Based on In Juence Functions 66
1.3e. The Relation between the Minimax Approach and the Approach Based on Influence Functions 73
1.3f. The Approach Based on Influence Functions as a Robustifed Likelihood Approach, and Its Relation to Various Statistical Schools 78
*1.4. Rejection of Outliers and Robust Statistics 82
1.4a. Why Rejection of Outliers? 82
1.4b. How Well Are Objective and Subjective Methods for the Rejection of Outliers Doing in the Context of Robust Estimation? 88
Exercises and Problems 97
2. ONE-DIMENSIONAL ESTIMATORS 104
2.0. An Introductory Example 104
2.1. The Influence Function 107
2.1a. Parametric Models, Estimators, and Functionals 107
2.1b. Definition and Properties of the Influence Function 109
2.1c. Robustness Measures Derived from the Influence Function 113
2.1d. Some Simple Examples 114
2.1e. Finite-Sample Versions 118
2.2. The Breakdown Point and Qualitative Robustness 122
2.2a. Global Reliability: The Breakdown Point 122
2.2b. Continuity and Qualitative Robustness 124
2.3. Classes of Estimators 126
2.3a. M-Estimators 126
2.3b. L-Estimators 134
2.3c. R-Estimators 136
2.3d. Other Types of Estimators: A, D, P, S, W 139
2.4. Optimally Bounding the Gross-Error Sensitivity 142
2.4a. The General Optimality Result 142
2.4b. M-Estimator 145
2.4c. L-Estimators 148
2.4d. R-Estimators 150
2.5. The Change-of-Variance Function 151
2.5a. Definitions 151
2.5b. B-Robustness versus V-Robustness 157
2.5c. The Most Robust Estimator 159
2.5d. Optimal Robust Estimators 160
2.5e. M-Estimators for Scale 165
*2.5f. Further Topics 170
2.6. Redescending M-Estimators 175
2.6a. Introduction 175
2.6b. Most Robust Estimators 180
2.6c. Optimal Robust Estimators 184
2.6d. Schematic Summary of Sections 2.5 and 2.6 194
*2.6e. Redescending M-Estimators for Scale 194
2.7. Relation with Huber\u2019s Minimax Approach 198
Exercises and Problems 204
3. ONE-DIMENSIONAL TESTS 213
3.1. Introduction 213
3.2. The Influence Function for Tests 215
3.2a. Definition of the Influence Function 215
3.2b. Properties of the Influence Function 220
3.2c. Relation with Level and Power 224
3.2d. Connection with Shift Estimators 228
3.3. Classes of Tests 230
3.3a The One-Sample Case 230
3.3b. The Two-Sample Case 232
3.4. Optimally Bounding the Gross-Error Sensitivity 235
3.5. Extending the Change-of-Variance Function to Tests 238
*3.6. Related Approaches 241
3.6a. Lambert\u2019s Approach 241
3.6b. Eplett\u2019s Approach 244
*3.7. M-Tests for a Simple Alternative 245
Exercises and Problems 247
4. MULTIDIMENSIONAL ESTIMATORS 251
4.1. Introduction 251
4.2. Concepts 252
4.2a. Influence Function 252
4.2b. Gross-Error Sensitivities 254
4.2c. M-Estimators 256
4.2d. Example: Location and Scale 258
4.3. Optimal Estimators 264
4.3a. The Unstandardized Case 264
4.3b. The Optimal B-Robust Estimators 269
4.3c. Existence and Uniqueness of the Optimal 蠄-Functions 272
4.3d How to Obtain Optimal Estimators 273
4.4. Partitioned Parameters 278
4.4a. Introduction: Location and Scale 278
*4.4b. Optimal Estimators 279
*4.5. Invariance 283
4.5a. Models Generated by Transformations 283
4.5b. Models and Invariance 283
4.5c. Equivariant Estimators 285
*4.6. Complements 286
4.6a. Admissible B-Robust Estimators 286
4.6b. Calculation of M-Estimates 289
Exercises and Problems 292
5. ESTIMATION OF COVARIANCE MATRICES AND MULTIVARIATE LOCATION 296
5.1. Introduction 296
5.2. The Model 297
5.2a. Definition 297
5.2b Scores 300
5.3. Equivariant Estimators 301
5.3a. Orthogonally Equivariant Vector Functions and d-Type Matrices 301
5.3b. General Results 306
5.3c. M-Estimators 309
5.4. Optimal and Most B-Robust Estimators 315
5.4a Full Parameter 315
5.4b. Partitioned Parameter 319
5.5. Breakdown Properties of Covariance Matrix Estimators 322
5.5a. Breakdown Point of M-Estimators 322
*5.5b. Breakdown at the Edge 325
5.5c. An Estimator with Breakdown Point 1/2 326
Exercises and Problems 329
6. LINEAR MODELS: ROBUST ESTIMATION 333
6.1. Introduction 333
6.1a. Overview 333
6.1b. The Model and the Classical Least-Squares Estimates 334
6.2. Huber-Estimators 337
6.3. M-Estimators for Linear Models 341
6.3a. Definition, Influence Function, and Sensitivities 341
6.3b. Most B-Robust and Optimal B-Robust Estimators 344
6.3c. The Change-of-Variance Function; Most V-Robust and Optimal V-Robust Estimators 349
6.4. Complements 354
6.4a. Breakdown Aspects 354
*6.4b. Asymptotic Behavior of Bounded Influence Estimators 357
6.4c. Computer Programs 361
*6.4d. Related Approaches 363
Exercises and Problems 364
7. LINEAR MODELS: ROBUST TESTING 368
7.1. Introduction 368
7.1a. Overview 368
7.lb. The Test Problem in Linear Models 369
7.2. A General Class of Tests for Linear Models 371
7.2a. Definition of 蟿Test 371
7.2b. Influence Function and Asymptotic Distribution 373
7.2c. Special Cases 380
7.3. Optimal Bounded Influence Tests 384
7.3a The pc-Test 384
7.3b The Optimal Mallows-Type Test 385
7.3c. The Optimal Test for the General M-Regression 386
7.3d. A Robust Procedure for Model Selection 392
*7.4. C(伪) \u2013Type Tests for Linear Models 393
7.4a. Definition of a C(伪) \u2013Type Test 394
7.4b. Influence Function and Asymptotic Power of C(伪)\u2013Type Tests 395
7.4c. Optimal Robust C(伪)\u2013Type Tests 399
7.4d Connection with an Asymptotically Minimax Test 399
7.5. Complements 402
*7.5a. Computation of Optimal 畏 Functions 402
*7.5b. Computation of the Asymptotic Distribution of the 蟿\u2013Test Statistic 403
*7.5c. Asymptotic Behavior of Direrent Tests for Simple Regression 404
7.5d. A Numerical Example 408
Exercises and Problems 411
8. COMPLEMENTS AND OUTLOOK 413
8.1. The Problem of Unsuspected Serial Correlations, or Violation of the Independence Assumption 413
8.la. Empirical Evidence for Semi-systematic Errors 413
8.1b. The Model of Self-similar Processes for Unsuspected Serial Correlations 415
8.1c. Some Consequences of the Model of Self-similar Processes 417
8.ld Estimation of the Long-Range Intensity of Serial Correlations 421
8.le. Some Further Problems of Robustness against Serial Correlations 422
8.2. Some Frequent Misunderstandings about Robust Statistics 423
8.2a. Some Common Objections against Huber\u2019s Minimax Approach 423
8.2b. \u201cRobust Statistics Is Not Necessary, Because ... \u201d 429
*8.2c. Some Details on Redescending Estimators 432
*8.2d. What Can Actually Be Estimated? 435
8.3. Robustness in Time Series 442
8.3a. Introduction 442
8.3b. The Influence Function for Time Series 443
8.3c. Other Robustness Problems in Time Series 448
*8.4. Some Special Topics Related to the Breakdown Point 448
8.4a. Most Robust (Median-Type) Estimators on the Real Line 448
8.4b. Special Structural Aspects of the Analysis of Variance 451
*8.5. Small-Sample Asymptotics 458
8.5a. Introduction 458
8.5b. Small-Sample Asymptotics for M-Estimators 459
8.5c. Further Applications 464
Exercises and Problems 464
REFERENCES 465
INDEX 491
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