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E-raamat: Robustness Theory and Application

(Murdoch University, Perth, Western Australia)
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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(26)
1.1 Introduction
1(1)
1.2 Probability Spaces and Distribution Functions
2(1)
1.3 Laws of Large Numbers
3(5)
1.3.1 Convergence in Probability and Almost Sure
3(1)
1.3.2 Expectation and Variance
4(1)
1.3.3 Statements of the Law of Large Numbers
4(1)
1.3.4 Some History and an Example
5(1)
1.3.5 Some More Asymptotic Theory and Application
6(2)
1.4 The Modus Operandi Related by Location Estimation
8(9)
1.5 Efficiency of Location Estimators
17(3)
1.6 Estimation of Location and Scale
20(7)
2 The Functional Approach
27(32)
2.1 Estimation and Conditions A
27(10)
2.2 Consistency
37(4)
2.3 Weak Continuity and Weak Convergence
41(3)
2.4 Frechet Differentiability
44(4)
2.5 The Influence Function
48(3)
2.6 Efficiency for Multivariate Parameters
51(1)
2.7 Other Approaches
52(7)
3 More Results on Differentiability
59(20)
3.1 Further Results on Frechet Differentiability
59(1)
3.2 M-Estimators: Their Introduction
59(11)
3.2.1 Non-Smooth Analysis and Conditions A'
61(4)
3.2.2 Existence and Uniqueness for Solutions of Equations
65(2)
3.2.3 Results for M-estimators with Non-Smooth Ψ
67(3)
3.3 Regression M-Estimators
70(3)
3.4 Stochastic Frechet Expansions and Further Considerations
73(1)
3.5 Locally Uniform Frechet Expansion
74(2)
3.6 Concluding Remarks
76(3)
4 Multiple Roots
79(20)
4.1 Introduction to Multiple Roots
79(1)
4.2 Asymptotics for Multiple Roots
80(2)
4.3 Consistency in the Face of Multiple Roots
82(17)
4.3.1 Preliminaries
83(9)
4.3.2 Asymptotic Properties of Roots and Tests
92(2)
4.3.3 Application of Asymptotic Theory
94(3)
4.3.4 Normal Mixtures and Conclusion
97(2)
5 Differentiability and Bias Reduction
99(14)
5.1 Differentiability, Bias Reduction, and Variance Estimation
99(9)
5.1.1 The Jackknife Bias and Variance Estimation
99(3)
5.1.2 Simple Location and Scale Bias Adjustments
102(3)
5.1.3 The Bootstrap
105(2)
5.1.4 The Choice to Jackknife or Bootstrap
107(1)
5.2 Further Results on the Newton Algorithm
108(5)
6 Minimum Distance Estimation and Mixture Estimation
113(34)
6.1 Minimum Distance Estimation and Revisiting Mixture Modeling
113(12)
6.2 The L2-Minimum Distance Estimator for Mixtures
125(10)
6.2.1 The L2-Estimator for Mixing Proportions
126(4)
6.2.2 The L2-Estimator for Switching Regressions
130(3)
6.2.3 An Example Application of Switching Regressions
133(2)
6.3 Other Minimum Distance Estimation Applications
135(12)
6.3.1 Mixtures of Exponential Distributions
136(3)
6.3.2 Gamma Distributions and Quality Assurance
139(8)
7 L-Estimates and Trimmed Likelihood Estimates
147(28)
7.1 A Preview of Estimation Using Order Statistics
147(5)
7.1.1 The Functional Form of L-Estimators of Location
150(2)
7.2 The Trimmed Likelihood Estimator
152(8)
7.2.1 LTS and Breakdown Point
154(2)
7.2.2 TLE Asymptotics for the Normal Distribution
156(4)
7.3 Adaptive Trimmed Likelihood and Identification of Outliers
160(3)
7.4 Adaptive Trimmed Likelihood in Regression
163(6)
7.5 What to do if n is Large?
169(6)
7.5.1 TLE Asymptotics for Location and Regression
170(5)
8 Trimmed Likelihood for Multivariate Data
175(6)
8.1 Identification of Multivariate Outliers
175(6)
9 Further Directions and Conclusion
181(6)
9.1 A Way Forward
181(6)
Appendix A Specific Proof of Theorem 2.1 187(2)
Appendix B Specific Calculations in Examples 4.1 and 4.2 189(4)
Appendix C Calculation of Moments in Example 4.2 193(2)
Bibliography 195(16)
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.
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