Robustness Theory and Application

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.

• Foreword xiPreface xvAcknowledgments xviiNotation xixAcronyms xxiAbout the Companion Website xxiii1 Introduction to Asymptotic Convergence 11.1 Introduction, 11.2 Probability Spaces and Distribution Functions, 21.3 Laws of Large Numbers, 31.3.1 Convergence in Probability and Almost Sure, 31.3.2 Expectation and Variance, 41.3.3 Statements of the Law of Large Numbers, 41.3.4 Some History and an Example, 51.3.5 Some More Asymptotic Theory and Application, 61.4 The Modus Operandi Related by Location Estimation, 81.5 Efficiency of Location Estimators, 171.6 Estimation of Location and Scale, 202 The Functional Approach 272.1 Estimation and Conditions A, 272.2 Consistency, 372.3 Weak Continuity and Weak Convergence, 412.4 Fréchet Differentiability, 442.5 The Influence Function, 482.6 Efficiency for Multivariate Parameters, 512.7 Other Approaches, 523 More Results on Differentiability 593.1 Further Results on Fréchet Differentiability, 593.2 M-Estimators: Their Introduction, 593.2.1 Non-Smooth Analysis and Conditions A′, 613.2.2 Existence and Uniqueness for Solutions of Equations, 653.2.3 Results for M-estimators with Non-Smooth Ψ, 673.3 Regression M-Estimators, 703.4 Stochastic Fréchet Expansions and Further Considerations, 733.5 Locally Uniform Fréchet Expansion, 743.6 Concluding Remarks, 764 Multiple Roots 794.1 Introduction to Multiple Roots, 794.2 Asymptotics for Multiple Roots, 804.3 Consistency in the Face of Multiple Roots, 824.3.1 Preliminaries, 834.3.2 Asymptotic Properties of Roots and Tests, 924.3.3 Application of Asymptotic Theory, 944.3.4 Normal Mixtures and Conclusion, 975 Differentiability and Bias Reduction 995.1 Differentiability, Bias Reduction, and Variance Estimation, 995.1.1 The Jackknife Bias and Variance Estimation, 995.1.2 Simple Location and Scale Bias Adjustments, 1025.1.3 The Bootstrap, 1055.1.4 The Choice to Jackknife or Bootstrap, 1075.2 Further Results on the Newton Algorithm, 1086 Minimum Distance Estimation and Mixture Estimation 1136.1 Minimum Distance Estimation and Revisiting Mixture Modeling, 1136.2 The L2-Minimum Distance Estimator for Mixtures, 1256.2.1 The L2-Estimator for Mixing Proportions, 1266.2.2 The L2-Estimator for Switching Regressions, 1306.2.3 An Example Application of Switching Regressions, 1336.3 Other Minimum Distance Estimation Applications, 1356.3.1 Mixtures of Exponential Distributions, 1366.3.2 Gamma Distributions and Quality Assurance, 1397 L-Estimates and Trimmed Likelihood Estimates 1477.1 A Preview of Estimation Using Order Statistics, 1477.1.1 The Functional Form of L-Estimators of Location, 1507.2 The Trimmed Likelihood Estimator, 1527.2.1 LTS and Breakdown Point, 1547.2.2 TLE Asymptotics for the Normal Distribution, 1567.3 Adaptive Trimmed Likelihood and Identification of Outliers, 1607.4 Adaptive Trimmed Likelihood in Regression, 1637.5 What to do if n is Large?, 1697.5.1 TLE Asymptotics for Location and Regression, 1708 Trimmed Likelihood for Multivariate Data 1758.1 Identification of Multivariate Outliers, 1759 Further Directions and Conclusion 1819.1 A Way Forward, 181Appendix A Specific Proof of Theorem 2.1 187Appendix B Specific Calculations in Examples 4.1 and 4.2 189Appendix C Calculation of Moments in Example 4.2 193Bibliography 195Index 211

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