Robust Statistics, 2nd Edition

by Peter J. Huber (formerly of ETH Zurich, Switzerland ); Elvezio M. Ronchetti (University of Geneva, Switzerland)

• ISBN13: 9780470434680
• ISBN10: 0470434686
• Format: eBook
• Copyright: 2009-02-01
• Publisher: Wiley

Summary

Robust Statistics, Second Edition includes four new chapters on the following topics: robust tests; small sample asymptotics; breakdown point; and Bayesian robustness. A new section on time series has also been included. The first edition of this book was the first systematic, book-length treatment of robust statistics. The book begins with a general introduction and the formal mathematical background behind qualitative and quantitative robustness. A solid foundation of robust statistics for both the theoretical and the applied statistician is provided. The book successfully reorganizes, summarizes, and extends information that has been available in part thus far. Concepts are stressed throughout rather than mathematical completeness, and selected numerical algorithms for computing robust estimates, as well as convergence proofs, are provided. Quantitative robustness information for a variety of estimates is contained within tables throughout.

Table of Contents

Preface

Preface to First Edition

Generalities

Why Robust Procedures?

What Should a Robust Procedure Achieve?

Qualitative Robustness

Quantitative Robustness

Infinitesimal Aspects

Optimal Robustness

Computation of Robust Estimates

Limitations to Robustness Theory

The Weak Topology and its Metrization

General Remarks

The Weak Topology

Lévy and Prohorov Metrics

The Bounded Lipschitz Metric

Fréechet and Gâteaux Derivatives

Hampel's Theorem

The Basic Types of Estimates

General Remarks

Maximum Likelihood Type Estimates (MEstimates)

Linear Combinations of Order Statistics (LEstimates)

Estimates Derived from Rank Tests (REstimates)

Asymptotically Efficient M, L, and REstimates

Asymptotic Minimax Theory for Estimating Location

General Remarks

Minimax Bias

Minimax Variance: Preliminaries

Distributions Minimizing Fisher Information

Determination of F0 by Variational Methods

Asymptotically Minimax MEstimates

On the Minimax Property for Land REstimates

Redescending MEstimates

Questions of Asymmetric Contamination

Scale Estimates

General Remarks

MEstimates of Scale

LEstimates of Scale

REstimates of Scale

Asymptotically Efficient Scale Estimates

Distributions Minimizing Fisher Information for Scale

Minimax Properties

Multiparameter Problems, in Particular Joint Estimation of Location and Scale

General Remarks

Consistency of MEstimates

Asymptotic Normality of MEstimates

Simultaneous MEstimates of Location and Scale

MEstimates with Preliminary Estimates of Scale

Quantitative Robustness of Joint Estimates of Location and Scale

The Computation of MEstimates of Scale

Studentizing

Regression

General Remarks

The Classical Linear Least Squares Case

Residuals and Outliers

Robustizing the Least Squares Approach

Asymptotics of Robust Regression Estimates

Conjectures and Empirical Results

Asymptotic Covariances and Their Estimation

Concomitant Scale Estimates

Computation of Regression MEstimates

The Fixed Carrier Case: what size h_i?

Analysis of Variance

L1estimates and Median Polish

Other Approaches to Robust Regression

Robust Covariance and Correlation Matrices

General Remarks

Estimation of Matrix Elements Through Robust Variances

Estimation of Matrix Elements Through Robust Correlation

An Affinely Equivariant Approach

Estimates Determined by Implicit Equations

Existence and Uniqueness of Solutions

Influence Functions and Qualitative Robustness

Consistency and Asymptotic Normality

Breakdown Point

Least Informative Distributions

Some Notes on Computation

Robustness of Design

General Remarks

Minimax Global Fit

Minimax Slope

Exact Finite Sample Results

General Remarks

Lower and Upper Probabilities and Capacities

Robust Tests

Sequential Tests

The NeymanPearson Lemma for 2Alternating Capacities

Estimates Derived From Tests

Minimax Interval Estimates

Finite Sample Breakdown Point

General Remarks

Definition and Examples

Infinitesimal Robustness and Breakdown

Malicious versus Stochastic Breakdown

Infinitesimal Robustness

General Remarks

Hampel's Infinitesimal Approach

Shrinking Neighborhoods

Robust Tests

General Remarks

Local Stability of a Test

Tests for General Parametric Models in the Multivariate Case

Robust Tests for Regression and Generalized Linear Models

Small Sample Asymptotics

General Remarks

Saddlepoint Approximation for the Mean

Saddlepoint Approximation of the Density of Mestimators

Tail Probabilities

Marginal Distributions

Saddlepoint Test

Relationship with Nonparametric Techniques

Bayesian Robustness

General Remarks

Disparate Data and Problems with the Prior

Maximum Likelihood and Bayes Estimates

Some Asymptotic Theory

Minimax Asymptotic Robustness Aspects

Nuisance Parameters

Why there is no Finite Sample Bayesian Robustness Theory

References

Index

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