Robust Statistics The Approach Based on Influence Functions

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

• ISBN13: 9780471735779
• ISBN10: 0471735779
• Edition: 1st
• Format: Paperback
• Copyright: 2005-04-06
• Publisher: Wiley-Interscience

Summary

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.

Author Biography

FRANK R. HAMPEL, PhD, is Professor of Statistics in the Department of Mathematics at the Swiss Federal Institute of Technology (ETH) Zurich, Switzerland.

ELVEZIO M. RONCHETTI, PhD, is Professor of Statistics in the Department of Econometrics at the University of Geneva in Switzerland.

PETER J. ROUSSEEUW, PhD, is Professor in the Department of Mathematics and Computer Science at the University of Antwerp in Belgium.

WERNER A. STAHEL, PhD, is Professor at the Swiss Federal Institute of Technology (ETH) Zurich, Switzerland.

Table of Contents

1. INTRODUCTION AND MOTIVATION	1	(77)
1.1. The Place and Aims of Robust Statistics	1	(17)
1.1a. What Is Robust Statistics?	1	(7)
1.1b. The Relation to Some Other Key Words in Statistics,	8	(3)
1.1c. The Aims of Robust Statistics,	11	(3)
1.1d. An Example,	14	(4)
1.2. Why Robust Statistics?	18	(16)
1.2a. The Role of Parametric Models,	18	(2)
1.2b. Types of Deviations from Parametric Models,	20	(5)
1.2c. The Frequency of Gross Errors,	25	(3)
1.2d. The Effects of Mild Deviations from a Parametric Model,	28	(3)
1.2e. How Necessary Are Robust Procedures?	31	(3)
1.3. The Main Approaches towards a Theory of Robustness	34	(22)
1.3a. Some Historical Notes,	34	(2)
1.3b. Huber's Minimax Approach for Robust Estimation,	36	(3)
1.3c. Huber's Second Approach to Robust Statistics via Robustified Likelihood Ratio Tests,	39	(1)
1.3d. The Approach Based on Influence Functions,	40	(7)
1.3e. The Relation between the Minimax Approach and the Approach Based on Influence Functions,	47	(5)
1.3f. The Approach Based on Influence Functions as a Robustified Likelihood Approach, and Its Relation to Various Statistical Schools,	52	(4)
1.4. Rejection of Outliers and Robust Statistics	56	(15)
1.4a. Why Rejection of Outliers?	56	(6)
1.4b. How Well Are Objective and Subjective Methods for the Rejection of Outliers Doing in the Context of Robust Estimation?	62	(9)
Exercises and Problems	71	(7)
2. ONE-DIMENSIONAL ESTIMATORS	78	(109)
2.0. An Introductory Example	78	(3)
2.1. The Influence Function	81	(15)
2.1a. Parametric Models, Estimators, and Functionals,	81	(2)
2.1b. Definition and Properties of the Influence Function,	83	(4)
2.1c. Robustness Measures Derived from the Influence Function,	87	(1)
2.1d. Some Simple Examples,	88	(4)
2.1e. Finite-Sample Versions,	92	(4)
2.2. The Breakdown Point and Qualitative Robustness	96	(4)
2.2a. Global Reliability: The Breakdown Point,	96	(2)
2.2b. Continuity and Qualitative Robustness,	98	(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