# Robust Statistics: Theory and Methods

# Robust Statistics: Theory and Methods

ISBN: 978-0-470-01094-5 June 2006 436 Pages

## Description

Classical statistical techniques fail to cope well with deviations from a standard distribution. Robust statistical methods take into account these deviations while estimating the parameters of parametric models, thus increasing the accuracy of the inference. Research into robust methods is flourishing, with new methods being developed and different applications considered.*Robust Statistics* sets out to explain the use of robust methods and their theoretical justification. It provides an up-to-date overview of the theory and practical application of the robust statistical methods in regression, multivariate analysis, generalized linear models and time series. This unique book:

- Enables the reader to select and use the most appropriate robust method for their particular statistical model.
- Features computational algorithms for the core methods.
- Covers regression methods for data mining applications.
- Includes examples with real data and applications using the S-Plus robust statistics library.
- Describes the theoretical and operational aspects of robust methods separately, so the reader can choose to focus on one or the other.
- Supported by a supplementary website featuring time-limited S-Plus download, along with datasets and S-Plus code to allow the reader to reproduce the examples given in the book.

*Robust Statistics* aims to stimulate the use of robust methods as a powerful tool to increase the reliability and accuracy of statistical modelling and data analysis. It is ideal for researchers, practitioners and graduate students of statistics, electrical, chemical and biochemical engineering, and computer vision. There is also much to benefit researchers from other sciences, such as biotechnology, who need to use robust statistical methods in their work.

**Preface.**

**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.**

*Journal of the American Statistical Association*, June 2008)

"…an original and valuable contribution…a source of inspiration for all those pursuing research in robust statistics." (*Mathematical Reviews*, 2007i)

"…a great book for graduate students as well as for applied scientists and data analysts." (*MAA Reviews*, February 14, 2007)