Summary

Robust Statistics is an introduction to the concepts, theory, and applications of the statistics of approximate parametric models. It gives a comprehensive account of the approach based on influence functions, which offers new insight into the robustness properties of existing procedures and provides a simple and flexible tool to derive new robust methods. The key concept, the influence function, describes the effect of an outlier on any locally linear statistical procedure. Related notions such as the breakdown point as a global measure of reliability and the change-of-variance function as a measure for the stability of the asymptotic variance are also covered in terms of general principles and applications.

Robust Statistics emphasizes ideas, heuristics and background rather than highly advanced mathematics. It includes a basic discussion of alternative approaches to robustness and the rejection of outliers.

Topics include:

• A general discussion of robustness and optimality problems within the framework of estimation of a single parameter.
• Unified treatment of estimators and tests, and univariate and multivariate problems.
• The estimation of a general multivariate parameter.
• Estimation of covariance matrices.
• Estimation and testing in multiple linear regression models.
• Some preliminary results on robustness in time series.
• The problem of unsuspected serial correlations.
• The relationship between the adopted approach and various less general approaches.
• Common misconceptions about robust statistics and an outlook on open problems.

The book addresses both theoretical and applied statisticians, without requiring previous knowledge of robustness. Its detailed explanations, practical examples, and numerous exercises make it accessible to a wide audience and encourage its use as a textbook. The researcher in statistics will find many hints to open problems and an extensive list of references.

Robust Statistics is an invaluable text/reference for anyone who wants a thorough understanding of the basic robustness notions and the results of the approach based on Influence Functions.

Table of contents

1. Introduction and Motivation
   1. The Place and Aims of Robust Statistics
   2. Why Robust Statistics?
   3. The Main Approaches towards a Theory of Robustness
   4. Rejection of Outliers and Robust Statistics
   Exercises and Problems
   
   
2. One-Dimensional Estimators
   1. An Introductory Example
   2. The Influence Function
   3. The Breakdown Point and Qualitative Robustness
   4. Classes of Estimators
   5. Optimally Bounding the Gross-Error Sensitivity
   6. The Change-of-Variance Function
   7. Redescending M-Estimators
   8. Relation with Huber's Minimax Approach
   Exercises and Problems
   
   
3. One-Dimensional Tests
   1. Introduction
   2. The Influence Function for Tests
   3. Classes of Tests
   4. Optimally Bounding the Gross-Error Sensitivity
   5. Extending the Change-of-Variance Function to Tests
   6. Related Approaches
   7. M-Tests for a Simple Alternative
   Exercises and Problems
   
   
4. Multidimensional Estimators
   1. Introduction
   2. Concepts
   3. Optimal Estimators
   4. Partitioned Parameters
   5. Invariance
   6. Complements
   Exercises and Problems
   
   
5. Estimation of Covariance Matrices and Multivariate Location
   1. Introduction
   2. The Model
   3. Equivariant Estimators
   4. Optimal and Most B-Robust Estimators
   5. Breakdown Properties of Covariance Matrix Estimators
   Exercises and Problems
   
   
6. Linear Models: Robust Estimation
   1. Introduction
   2. Huber-Estimators
   3. M-Estimators for Linear Models
   4. Complements
   Exercises and Problems
   
   
7. Linear Models: Robust Testing
   1. Introduction
   2. A General Class of Tests for Linear Models
   3. Optimal Bounded Influence Tests
   4. C(α)-Type Tests for Linear Models
   5. Complements
   Exercises and Problems
   
   
8. Complements and Outlook
   1. The Problem of Unsuspected Serial Correlations, or Violation of the Independence Assumption
   2. Some Frequent Misunderstandings about Robust Statistics
   3. Robustness in Time Series
   4. Some Special Topics Related to the Breakdown Point
   5. Small-Sample Asymptotics
   Exercises and Problems
   
   

Book details

Wiley-Interscience, New York (Series in Probability and Mathematical Statistics), 502 pages.
ISBN 0-471-82921-8.
Second printing also as paperback (ISBN 0-471-63238-4), 14 reviews.
Russian translation (1989), Mir, Moscow.