Robust Statistics The Approach Based on Influence Functions

, by Hampel, Frank R.; Ronchetti, Elvezio M.; Rousseeuw, Peter J.; Stahel, Werner A.

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ISBN: 9780471735779 | 0471735779
Cover: Paperback
Copyright: 4/6/2005

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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.Robust Statistics is a leading-edge resource suitable for use both as a textbook or a reference on robust statistics for all practitioners and students.

1. INTRODUCTION AND MOTIVATION

(77)

1.1. The Place and Aims of Robust Statistics

(17)

1.1a. What Is Robust Statistics?

(7)

1.1b. The Relation to Some Other Key Words in Statistics,

(3)

1.1c. The Aims of Robust Statistics,

(3)

1.1d. An Example,

(4)

1.2. Why Robust Statistics?

(16)

1.2a. The Role of Parametric Models,

(2)

1.2b. Types of Deviations from Parametric Models,

(5)

1.2c. The Frequency of Gross Errors,

(3)

1.2d. The Effects of Mild Deviations from a Parametric Model,

(3)

1.2e. How Necessary Are Robust Procedures?

(3)

1.3. The Main Approaches towards a Theory of Robustness

(22)

1.3a. Some Historical Notes,

(2)

1.3b. Huber's Minimax Approach for Robust Estimation,

(3)

1.3c. Huber's Second Approach to Robust Statistics via Robustified Likelihood Ratio Tests,

(1)

1.3d. The Approach Based on Influence Functions,

(7)

1.3e. The Relation between the Minimax Approach and the Approach Based on Influence Functions,

(5)

1.3f. The Approach Based on Influence Functions as a Robustified Likelihood Approach, and Its Relation to Various Statistical Schools,

(4)

1.4. Rejection of Outliers and Robust Statistics

(15)

1.4a. Why Rejection of Outliers?

(6)

1.4b. How Well Are Objective and Subjective Methods for the Rejection of Outliers Doing in the Context of Robust Estimation?

(9)

Exercises and Problems

(7)

2. ONE-DIMENSIONAL ESTIMATORS

(109)

2.0. An Introductory Example

(3)

2.1. The Influence Function

(15)

2.1a. Parametric Models, Estimators, and Functionals,

(2)

2.1b. Definition and Properties of the Influence Function,

(4)

2.1c. Robustness Measures Derived from the Influence Function,

(1)

2.1d. Some Simple Examples,

(4)

2.1e. Finite-Sample Versions,

(4)

2.2. The Breakdown Point and Qualitative Robustness

(4)

2.2a. Global Reliability: The Breakdown Point,

(2)

2.2b. Continuity and Qualitative Robustness,

(2)

2.3. Classes of Estimators

100

(16)

2.3a. M-Estimators,

100

(8)

2.3b. L-Estimators,

108

(2)

2.3c. R-Estimators,

110

(3)

2.3d. Other Types of Estimators: A, D, P, S, W,

113

(3)

2.4. Optimally Bounding the Gross-Error Sensitivity

116

(9)

2.4a. The General Optimality Result,

116

(3)

2.4b. M-Estimators,

119

(3)

2.4c. L-Estimators,

122

(2)

2.4d. R-Estimators,

124

(1)

2.5. The Change-of-Variance Function

125

(24)

2.5a. Definitions,

125

(6)

2.5b. B-Robustness versus V-Robustness,

131

(2)

2.5c. The Most Robust Estimator,

133

(1)

2.5d. Optimal Robust Estimators,

134

(5)

2.5e. M-Estimators for Scale,

139

(5)

2.5f. Further Topics,

144

(5)

2.6. Redescending M-Estimators

149

(23)

2.6a. Introduction,

149

(5)

2.6b. Most Robust Estimators,

154

(4)

2.6c. Optimal Robust Estimators,

158

(10)

2.6d. Schematic Summary of Sections 2.5 and 2.6,

168

(1)

2.6e. Redescending M-Estimators for Scale,

168

(4)

2.7. Relation with Huber's Minimax Approach

172

(6)

Exercises and Problems

178

(9)

3. ONE-DIMENSIONAL TESTS

187

(38)

3.1. Introduction

187

(2)

3.2. The Influence Function for Tests

189

(15)

3.2a. Definition of the Influence Function,

189

(5)

3.2b. Properties of the Influence Function,

194

(4)

3.2c. Relation with Level and Power,

198

(4)

3.2d Connection with Shift Estimators,

202

(2)

3.3. Classes of Tests

204

(5)

3.3a. The One-Sample Case,

204

(2)

3.3b. The Two-Sample Case,

206

(3)

3.4. Optimally Bounding the Gross-Error Sensitivity

209

(3)

3.5. Extending the Change-of-Variance Function to Tests

212

(3)

3.6. Related Approaches

215

(4)

3.6a. Lambert's Approach,

215

(3)

3.6b. Eplett's Approach,

218

(1)

3.7. M-Tests for a Simple Alternative

219

(2)

Exercises and Problems

221

(4)

4. MULTIDIMENSIONAL ESTIMATORS

225

(45)

4.1. Introduction

225

(1)

4.2. Concepts

226

(12)

4.2a. Influence Function,

226

(2)

4.2b. Gross-Error Sensitivities,

228

(2)

4.2c. M-Estimators,

230

(2)

4.2d. Example: Location and Scale,

232

(6)

4.3. Optimal Estimators

238

(14)

4.3a. The Unstandardized Case,

238

(5)

4.3b. The Optimal B-Robust Estimators,

243

(3)

4.3c. Existence and Uniqueness of the Optimal ψ-Functions,

246

(1)

4.3d. How to Obtain Optimal Estimators,

247

(5)

4.4. Partitioned Parameters

252

(5)

4.4a. Introduction: Location and Scale,

252

(1)

4.4b. Optimal Estimators,

253

(4)

4.5. Invariance

257

(3)

4.5a. Models Generated by Transformations,

257

(1)

4.5b. Models and Invariance,

257

(2)

4.5c. Equivariant Estimators,

259

(1)

4.6. Complements

260

(6)

4.6a. Admissible B-Robust Estimators,

260

(3)

4.6b. Calculation of M-Estimates,

263

(3)

Exercises and Problems

266

(4)

5. ESTIMATION OF COVARIANCE MATRICES AND MULTIVARIATE LOCATION

270

(37)

5.1. Introduction

270

(1)

5.2. The Model

271

(4)

5.2a. Definition,

271

(3)

5.2b Scores,

274

(1)

5.3. Equivariant Estimators

275

(14)

5.3a. Orthogonally Equivariant Vector Functions and d-Type Matrices,

275

(5)

5.3b. General Results,

280

(3)

5.3c. M-Estimators,

283

(6)

5.4. Optimal and Most B-Robust Estimators

289

(7)

5.4a. Full Parameter,

289

(4)

5.4b. Partitioned Parameter,

293

(3)

5.5. Breakdown Properties of Covariance Matrix Estimators

296

(7)

5.5a. Breakdown Point of M-Estimators,

296

(3)

5.5b. Breakdown at the Edge,

299

(1)

5.5c. An Estimator with Breakdown Point ½,

300

(3)

Exercises and Problems

303

(4)

6. LINEAR MODELS: ROBUST ESTIMATION

307

(35)

6.1. Introduction

307

(4)

6.1a. Overview,

307

(1)

6.1b. The Model and the Classical Least-Squares Estimates,

308

(3)

6.2. Huber-Estimators

311

(4)

6.3. M-Estimators for Linear Models

315

(13)

6.3a. Definition, Influence Function, and Sensitivities,

315

(3)

6.3b. Most B-Robust and Optimal B-Robust Estimators,

318

(5)

6.3c. The Change-of-Variance Function; Most V-Robust and Optimal V-Robust Estimators,

323

(5)

6.4. Complements

328

(10)

6.4a. Breakdown Aspects,

328

(3)

6.4b. Asymptotic Behavior of Bounded Influence Estimators,

331

(4)

6.4c. Computer Programs,

335

(2)

6.4d. Related Approaches,

337

(1)

Exercises and Problems

338

(4)

7. LINEAR MODELS: ROBUST TESTING

342

(45)

7.1. Introduction

342

(3)

7.1a. Overview,

342

(1)

7.1b. The Test Problem in Linear Models,

343

(2)

7.2. A General Class of Tests for Linear Models

345

(13)

7.2a. Definition of τ-Test,

345

(2)

7.2b. Influence Function and Asymptotic Distribution,

347

(7)

7.2c. Special Cases,

354

(4)

7.3. Optimal Bounded Influence Tests

358

(9)

7.3a. The ρc-Test,

358

(1)

7.3b The Optimal Mallows-Type Test,

359

(1)

7.3c. The Optimal Test for the General M-Regression,

360

(6)

7.3d. A Robust Procedure for Model Selection,

366

(1)

7.4. C(α)-Type Tests for Linear Models

367

(9)

7.4a. Definition of a C(α)-Type Test,

368

(1)

7.4b. Influence Function and Asymptotic Power of C(α)-Type Tests,

369

(4)

7.4c. Optimal Robust C(&alapha;)-Type Tests,

373

(1)

7.4d. Connection with an Asymptotically Minimax Test,

373

(3)

7.5. Complements

376

(9)

7.5a. Computation of Optimal n Functions,

376

(1)

7.5b. Computation of the Asymptotic Distribution of the τ-Test Statistic,

377

(1)

7.5c. Asymptotic Behavior of Different Tests for Simple Regression,

378

(5)

7.5d. A Numerical Example,

383

(2)

Exercises and Problems

385

(2)

8. COMPLEMENTS AND OUTLOOK

387

(52)

8.1. The Problem of Unsuspected Serial Correlations, or Violation of the Independence Assumption

387

(10)

8.1a. Empirical Evidence for Semi-systematic Errors,

387

(2)

8.1b. The Model of Self-Similar Processes for Unsuspected Serial Correlations,

389

(2)

8.1c. Some Consequences of the Model of Self-Similar Processes,

391

(4)

8.1d. Estimation of the Long-Range Intensity of Serial Correlations,

395

(1)

8.1e. Some Further Problems of Robustness against Serial Correlations,

396

(1)

8.2. Some Frequent Misunderstandings about Robust Statistics

397

(19)

8.2a. Some Common Objections against Huber's Minimax Approach,

397

(6)

8.2b. "Robust Statistics Is Not Necessary, Because...',

403

(3)

8.2c. Some Details on Redescending Estimators,

406

(3)

8.2d. What Can Actually Be Estimated?

409

(7)

8.3. Robustness in Time Series

416

(6)

8.3a. Introduction,

416

(1)

8.3b. The Influence Function for Time Series,

417

(5)

8.3c. Other Robustness Problems in Time Series,

422

(1)

8.4. Some Special Topics Related to the Breakdown Point

422

(10)

8.4a. Most Robust (Median-Type) Estimators on the Real Line,

422

(3)

8.4b. Special Structural Aspects of the Analysis of Variance,

425

(7)

8.5. Small-Sample Asymptotics

432

(6)

8.5a. Introduction,

432

(1)

8.5b. Small-Sample Asymptotics for M-Estimators,

433

(5)

8.5c. Further Applications,

438

(1)

Exercises and Problems

438

(1)

REFERENCES

439

(26)

INDEX

465

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Robust Statistics The Approach Based on Influence Functions

Summary

Author Biography

Table of Contents

Supplemental Materials

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Robust Statistics The Approach Based on Influence Functions

Summary

Author Biography

Table of Contents

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