Robustness Theory and Application

A preeminent expert in the field explores new and exciting methodologies in the ever–growing field of robust statistics

Used to develop data analytical methods, which are resistant to outlying observations in the data, while capable of detecting outliers, robust statistics is extremely useful for solving an array of common problems, such as estimating location, scale, and regression parameters. Written by an internationally recognized expert in the field of robust statistics, this book addresses a range of well–established techniques while exploring, in depth, new and exciting methodologies. Local robustness and global robustness are discussed, and problems of non–identifiability and adaptive estimation are considered. Rather than attempt an exhaustive investigation of robustness, the author provides readers with a timely review of many of the most important problems in statistical inference involving robust estimation, along with a brief look at confidence intervals for location. Throughout, the author meticulously links research in maximum likelihood estimation with the more general M–estimation methodology. Specific applications and R and some MATLAB subroutines with accompanying data sets available both in the text and online are employed wherever appropriate.

Providing invaluable insights and guidance, Robustness Theory and Application:

Offers a balanced presentation of theory and applications within each topic–specific discussion Features solved examples throughout which help clarify complex and/or difficult concepts Meticulously links research in maximum likelihood type estimation with the more general M–estimation methodology Delves into new methodologies which have been developed over the past decade without stinting on coverage of "tried–and–true" methodologies Includes R and some MATLAB subroutines with accompanying data sets, which help illustrate the power of the methods described

Robustness Theory and Application is an important resource for all statisticians interested in the topic of robust statistics. This book encompasses both past and present research, making it a valuable supplemental text for graduate–level courses in robustness.

Foreword xi

Preface xv

Acknowledgments xvii

Notation xix

Acronyms xxi

About the Companion Website xxiii

1 Introduction to Asymptotic Convergence 1

1.1 Introduction, 1

1.2 Probability Spaces and Distribution Functions, 2

1.3 Laws of Large Numbers, 3

1.3.1 Convergence in Probability and Almost Sure, 3

1.3.2 Expectation and Variance, 4

1.3.3 Statements of the Law of Large Numbers, 4

1.3.4 Some History and an Example, 5

1.3.5 Some More Asymptotic Theory and Application, 6

1.4 The Modus Operandi Related by Location Estimation, 8

1.5 Efficiency of Location Estimators, 17

1.6 Estimation of Location and Scale, 20

2 The Functional Approach 27

2.1 Estimation and Conditions A, 27

2.2 Consistency, 37

2.3 Weak Continuity and Weak Convergence, 41

2.4 Fréchet Differentiability, 44

2.5 The Influence Function, 48

2.6 Efficiency for Multivariate Parameters, 51

2.7 Other Approaches, 52

3 More Results on Differentiability 59

3.1 Further Results on Fréchet Differentiability, 59

3.2 M–Estimators: Their Introduction, 59

3.2.1 Non–Smooth Analysis and Conditions A , 61

3.2.2 Existence and Uniqueness for Solutions of Equations, 65

3.2.3 Results for M–estimators with Non–Smooth , 67

3.3 Regression M–Estimators, 70

3.4 Stochastic Fréchet Expansions and Further Considerations, 73

3.5 Locally Uniform Fréchet Expansion, 74

3.6 Concluding Remarks, 76

4 Multiple Roots 79

4.1 Introduction to Multiple Roots, 79

4.2 Asymptotics for Multiple Roots, 80

4.3 Consistency in the Face of Multiple Roots, 82

4.3.1 Preliminaries, 83

4.3.2 Asymptotic Properties of Roots and Tests, 92

4.3.3 Application of Asymptotic Theory, 94

4.3.4 Normal Mixtures and Conclusion, 97

5 Differentiability and Bias Reduction 99

5.1 Differentiability, Bias Reduction, and Variance Estimation, 99

5.1.1 The Jackknife Bias and Variance Estimation, 99

5.1.2 Simple Location and Scale Bias Adjustments, 102

5.1.3 The Bootstrap, 105

5.1.4 The Choice to Jackknife or Bootstrap, 107

5.2 Further Results on the Newton Algorithm, 108

6 Minimum Distance Estimation and Mixture Estimation 113

6.1 Minimum Distance Estimation and Revisiting Mixture Modeling, 113

6.2 The L2–Minimum Distance Estimator for Mixtures, 125

6.2.1 The L2–Estimator for Mixing Proportions, 126

6.2.2 The L2–Estimator for Switching Regressions, 130

6.2.3 An Example Application of Switching Regressions, 133

6.3 Other Minimum Distance Estimation Applications, 135

6.3.1 Mixtures of Exponential Distributions, 136

6.3.2 Gamma Distributions and Quality Assurance, 139

7 L–Estimates and Trimmed Likelihood Estimates 147

7.1 A Preview of Estimation Using Order Statistics, 147

7.1.1 The Functional Form of L–Estimators of Location, 150

7.2 The Trimmed Likelihood Estimator, 152

7.2.1 LTS and Breakdown Point, 154

7.2.2 TLE Asymptotics for the Normal Distribution, 156

7.3 Adaptive Trimmed Likelihood and Identification of Outliers, 160

7.4 Adaptive Trimmed Likelihood in Regression, 163

7.5 What to do if n is Large?, 169

7.5.1 TLE Asymptotics for Location and Regression, 170

8 Trimmed Likelihood for Multivariate Data 175

8.1 Identification of Multivariate Outliers, 175

9 Further Directions and Conclusion 181

9.1 A Way Forward, 181

Appendix A Specific Proof of Theorem 2.1 187

Appendix B Specific Calculations in Examples 4.1 and 4.2 189

Appendix C Calculation of Moments in Example 4.2 193

Bibliography 195

Index 211

Brenton R. Clarke, PhD is an experienced academic in Mathematics and Statistics at Murdoch University, Perth, WA, Australia. A former president of the Western Australian Branch of the Statistical Society of Australia, Dr. Clarke has published numerous journal articles in his areas of research interest, which include linear models, robust statistics, and time series analysis.