Robust Statistics: Theory and Methods

1. Introduction.

1.1 Classical and robust approaches to statistics.

1.2 Mean and standard deviation.

1.3 The “three-sigma edit” rule.

1.4 Linear regression.

1.5 Correlation coefficients.

1.6 Other parametric models.

1.7 Problems.

2. Location and Scale.

2.1 The location model.

2.2 M-estimates of location.

2.3 Trimmed means.

2.4 Dispersion estimates.

2.5 M-estimates of scale.

2.6 M-estimates of location with unknown dispersion.

2.7 Numerical computation of M-estimates.

2.8 Robust confidence intervals and tests.

2.9 Appendix: proofs and complements.

2.10 Problems.

3. Measuring Robustness.

3.1 The influence function.

3.2 The breakdown point.

3.3 Maximum asymptotic bias.

3.4 Balancing robustness and efficiency.

3.5 *“Optimal” robustness.

3.6 Multidimensional parameters.

3.7 *Estimates as functionals.

3.8 Appendix: proofs of results.

3.9 Problems.

4 Linear Regression 1.

4.1 Introduction.

4.2 Review of the LS method.

4.3 Classical methods for outlier detection.

4.4 Regression M-estimates.

4.5 Numerical computation of monotone M-estimates.

4.6 Breakdown point of monotone regression estimates.

4.7 Robust tests for linear hypothesis.

4.8 *Regression quantiles.

4.9 Appendix: proofs and complements.

4.10 Problems.

5 Linear Regression 2.

5.1 Introduction.

5.2 The linear model with random predictors 118

5.3 M-estimates with a bounded ρ-function.

5.4 Properties of M-estimates with a bounded ρ-function.

5.5 MM-estimates.

5.6 Estimates based on a robust residual scale.

5.7 Numerical computation of estimates based on robust scales.

5.8 Robust confidence intervals and tests for M-estimates.

5.9 Balancing robustness and efficiency.

5.10 The exact fit property.

5.11 Generalized M-estimates.

5.12 Selection of variables.

5.13 Heteroskedastic errors.

5.14 *Other estimates.

5.15 Models with numeric and categorical predictors.

5.16 *Appendix: proofs and complements.

5.17 Problems.

6. Multivariate Analysis.

6.1 Introduction.

6.2 Breakdown and efficiency of multivariate estimates.

6.3 M-estimates.

6.4 Estimates based on a robust scale.

6.5 The Stahel–Donoho estimate.

6.6 Asymptotic bias.

6.7 Numerical computation of multivariate estimates.

6.8 Comparing estimates.

6.9 Faster robust dispersion matrix estimates.

6.10 Robust principal components.

6.11 *Other estimates of location and dispersion.

6.12 Appendix: proofs and complements.

6.13 Problems.

7. Generalized Linear Models.

7.1 Logistic regression.

7.2 Robust estimates for the logistic model.

7.3 Generalized linear models.

7.4 Problems.

8. Time Series.

8.1 Time series outliers and their impact.

8.2 Classical estimates for AR models.

8.3 Classical estimates for ARMA models.

8.4 M-estimates of ARMA models.

8.5 Generalized M-estimates.

8.6 Robust AR estimation using robust filters.

8.7 Robust model identification.

8.8 Robust ARMA model estimation using robust filters.

8.9 ARIMA and SARIMA models.

8.10 Detecting time series outliers and level shifts.

8.11 Robustness measures for time series.

8.12 Other approaches for ARMA models.

8.13 High-efficiency robust location estimates.

8.14 Robust spectral density estimation.

8.15 Appendix A: heuristic derivation of the asymptotic distribution of M-estimates for ARMA models.

8.16 Appendix B: robust filter covariance recursions.

8.17 Appendix C: ARMA model state-space representation.

8.18 Problems.

9. Numerical Algorithms.

9.1 Regression M-estimates.

9.2 Regression S-estimates.

9.3 The LTS-estimate.

9.4 Scale M-estimates.

9.5 Multivariate M-estimates.

9.6 Multivariate S-estimates.

10. Asymptotic Theory of M-estimates.

10.1 Existence and uniqueness of solutions.

10.2 Consistency.

10.3 Asymptotic normality.

10.4 Convergence of the SC to the IF.

10.5 M-estimates of several parameters.

10.6 Location M-estimates with preliminary scale.

10.7 Trimmed means.

10.8 Optimality of the MLE.

10.9 Regression M-estimates.

10.10 Nonexistence of moments of the sample median.

10.11 Problems.

11. Robust Methods in S-Plus.

11.1 Location M-estimates: function Mestimate.

11.2 Robust regression.

11.3 Robust dispersion matrices.

11.4 Principal components.

11.5 Generalized linear models.

11.6 Time series.

11.7 Public-domain software for robust methods.

12. Description of Data Sets.

Bibliography.

Index.