Robust and Multivariate Statistical Methods: Festschrift in Honor of David E. Tyler

Description

This book presents recent developments in multivariate and robust statistical methods. Featuring contributions by leading experts in the field it covers various topics, including multivariate and high-dimensional methods, time series, graphical models, robust estimation, supervised learning and normal extremes. It will appeal to statistics and data science researchers, PhD students and practitioners who are interested in modern multivariate and robust statistics. The book is dedicated to David E. Tyler on the occasion of his pending retirement and also includes a review contribution on the popular Tyler’s shape matrix. Foreword Preface Acknowledgements Contents List of Contributors Part I About David E. Tyler's Publications An Analysis of David E. Tyler's Publication and Coauthor Network 1 Introduction 2 David Tyler's Coauthor Network 3 David Tyler's Influence 4 Concluding Remarks Appendices A.1 Publications of David E Tyler A.2 R Packages of David E Tyler References A Review of Tyler's Shape Matrix and Its Extensions 1 Introduction 2 Definition and Statistical Properties 3 Extensions 3.1 Joint Estimation of Location and Tyler's Shape Matrix: The Hettmansperger–Randles Estimators 3.2 The Symmetrized Variant of Tyler's Shape Matrix: Dümbgen's Estimator 3.3 Estimation Under Missing Data 3.4 Structured Tyler's Shape Estimation 3.5 Regularized Estimators 4 Discussion References Part II Multivariate Theory and Methods On the Asymptotic Behavior of the Leading Eigenvector of Tyler's Shape Estimator Under Weak Identifiability 1 Introduction 2 Tyler's Estimator of Shape Under Weak Identifiability 3 Asymptotic Behavior of Tyler's Leading Eigenvector Under Weak Identifiability 4 Numerical Illustration Appendix References On Minimax Shrinkage Estimation with Variable Selection 1 Introduction 2 Results for the Normal Case, Known Scale 3 Scale Mixtures of Normal Distributions 4 Spherically Symmetric Distributions with Residual 5 A Simulation Study 6 Summary and Conclusion References On the Finite-Sample Performance of Measure-Transportation-Based Multivariate Rank Tests 1 Introduction 1.1 David Tyler, Beyond Affine Equivariance and Elliptical Symmetry 1.2 Ordering the Real Space in Dimension d≥2 2 Center-Outward Ranks and Signs 2.1 Measure Transportation-Based Concepts of Distribution and Quantile Functions 2.2 Multivariate Ranks and Signs 2.3 Distribution-Free Tests Based on Center-Outward Ranks and Signs 2.3.1 Score Functions 2.3.2 Test Statistics 3 Finite-Sample Performance: Two-Sample Location Simulations 3.1 Halton Sequences on the Cube and the Sphere ((Gii) and (Giv) Grids) 3.2 Factorization of n ((Gi) and (Giii) Grids) 3.3 Simulations 4 Wilcoxon-Type Tests 4.1 The Bivariate Case 4.1.1 Spherical Gaussian Samples 4.1.2 Nonspherical Gaussian Samples 4.1.3 Samples with Independent Cauchy Marginals 4.1.4 Spherical Cauchy Samples 4.1.5 ``Banana-Shaped'' Samples 4.2 Wilcoxon-Type Statistics in Dimension d=5 4.2.1 Spherical Gaussian Samples 4.2.2 Nonspherical Gaussian Samples 4.2.3 Samples with Independent Cauchy Marginals 4.3 Wilcoxon-Type Statistics in Dimension d=30 4.3.1 Spherical Gaussian Samples 4.3.2 Nonspherical Gaussian Samples 4.3.3 Samples with Independent Cauchy Marginals 4.4 Wilcoxon-Type Statistics in Dimension d=100 4.4.1 Spherical Gaussian Samples 4.4.2 Nonspherical Gaussian Samples 4.4.3 Samples with Independent Cauchy Marginals 5 van der Waerden-Type Tests 5.1 Bivariate Case 5.1.1 Spherical Gaussian Samples 5.1.2 Nonspherical Gaussian Samples 5.1.3 Samples with Independent Cauchy Marginals 5.1.4 Spherical Cauchy Samples 5.1.5 ``Banana-Shaped'' Samples 5.2 van der Waerden-Type Statistics in Dimension d=5 5.2.1 Spherical Gaussian Samples 5.2.2 Nonspherical Gaussian Samples 5.2.3 Samples with Independent Cauchy Marginals 5.3 van der Waerden-Type Statistics in Dimension d=30 5.3.1 Spherical Gaussian Samples 5.3.2 Nonspherical Gaussian Samples 5.3.3 Samples with Independent Cauchy Marginals 5.4 van der Waerden-Type Statistics in Dimension d=100 5.4.1 Spherical Gaussian Samples 5.4.2 Nonspherical Gaussian Samples 5.4.3 Samples with Independent Cauchy Marginals 6 Conclusions References Refining Invariant Coordinate Selection via Local Projection Pursuit 1 Projection Pursuit 2 Invariant Coordinate Selection as a Starting Point 3 Estimation of Entropy 4 Local Optimization 5 The Complete Procedure(s) 6 Numerical Examples References Directional Distributions and the Half-Angle Principle 1 Introduction 2 Basic Operations on the Circle 3 Transformations of Distributions on the Circle 4 Basic Operations on the Sphere 5 Projections from the Sphere to Euclidean Space 6 The ACG Distribution on the Sphere 6.1 Review of Quadratic Forms in the Multivariate Normal Distribution 6.2 Basic Properties of the ACG Distribution 6.3 ACG Distribution Under Gnomonic Projection 7 The Spherical Cauchy Distribution 8 Transformation Groups on the Sphere 9 Parameterizations and Motivations for the Wrapped Cauchy Distribution on S1 References Part III Robust Theory and Methods Power M-Estimators for Location and Scatter 1 Motivation 2 Prerequisites 3 Power M-Estimators for Location and Scatter 3.1 ML-Estimation 3.2 M-Estimation 3.3 Main Result 4 Asymptotic Distributions 4.1 Theoretical Results 4.2 A Simple Application 5 Proofs References On Robust Estimators of a Sphericity Measure in High Dimension 1 Introduction 2 On the Role of Sphericity on the Accuracy of SCM in High Dimension 3 Sphericity Estimator Based on the Sample Covariance Matrix 4 Sphericity Estimator Based on the Spatial Sign Covariance Matrix 5 Sphericity Estimators Based on M-Estimators of Scatter 6 Simulation Studies 7 Conclusions References Detecting Outliers in Compositional Data Using Invariant Coordinate Selection 1 Introduction 2 Reminder About ICS and Outlier Detection 2.1 Scatter Matrices 2.2 ICS Principle 2.3 ICS for Outlier Detection 3 Reminder About Compositional Data Analysis 4 Multivariate Tools for Compositional Data 4.1 Algebra of Endomorphisms of the Simplex and Eigendecomposition 4.2 One-Step M-Scatter Functionals of a Compositional Random Vector 4.3 Elliptical Distribution in the Simplex 5 ICS for Compositional Data 5.1 ICS in Coordinate Space 5.2 ICS in the Simplex 5.3 Reconstruction Formula 6 Examples of Application 6.1 Toy Examples 6.2 Market Shares Example 7 Conclusion Appendix References Robust Forecasting of Multiple Time Series with One-Sided Dynamic Principal Components 1 Introduction 2 One-Sided Dynamic Principal Components 3 Robust One-Sided Dynamic Principal Components 4 Computing Algorithm for the S-ODPC 5 Forecasting Using the S-ODPC 6 Selecting the Number of Lags and the Number of Components 6.1 Selection Using an Information Criterion 6.2 Selection Using Robust Cross-validation 7 Asymptotic Behavior of the S-ODPC in Factor Models 8 Simulation Results 9 Example with a Real Data Set 10 Conclusions Appendix Derivation of the Estimating Equations Proof of Theorem 1 References Robust and Sparse Estimation of Graphical Models Based on Multivariate Winsorization 1 Introduction 2 Outliers in High-Dimensional Data 3 Robust Lasso for Precision Matrices 3.1 Plug-in Strategy 3.2 Adjusted Multivariate Winsorization 4 Simulation Experiment and Numerical Results 4.1 Simulation Settings Precision Matrix Models Contamination Scenarios Precision Matrix Estimators Estimation Performance Evaluation 4.2 Estimation and Graph Recovery Performances 4.3 Timing Comparisons 5 Real Data Example 6 Concluding Remarks References Robustly Fitting Gaussian Graphical Models—the R Package robFitConGraph 1 Introduction 1.1 Gaussian Graphical Modeling 1.2 Robustness 2 A Case Study: Music Performance Anxiety 2.1 Inferential Analysis: MPA and Social Anxiety 2.2 Explorative Analysis 2.3 The Classical Analysis 3 Background and Theory 3.1 The Constant σ1 3.2 The Direct vs. the Plug-in Estimate 3.3 Ellipticity vs. Normality Technical Appendix References Robust Estimation of General Linear Mixed Effects Models 1 Introduction 2 The Model and Classical Estimation 3 The Robust Scoring Equations Estimator 3.1 Estimation for Diagonal Vb 3.2 Estimation for Block Diagonal Vb 3.3 Computation 3.4 Choices of ψ and w 3.5 Robust Tests 4 Properties of the Robust Scoring Equations Estimator 4.1 Sensitivity Curves 4.2 Efficiency and Robustness, Diagonal Case 4.3 Coverage Probabilities, Diagonal Case 4.4 Efficiency and Robustness, Block-Diagonal Case 5 Examples 5.1 Penicillin Data 5.2 Sleep Study 6 Conclusions Appendix Linear Approximation of Estimated Quantities Covariance Matrices Refined Design Adaptive Scale References Asymptotic Behaviour of Penalized Robust Estimators in Logistic Regression When Dimension Increases 1 Introduction 2 Preliminaries: Robust Penalized Estimators 2.1 Assumptions 3 Consistency and Rates of Convergence 4 Variable Selection and Asymptotic Distribution 5 General Comments Appendix 1: Proofs of Remark 4 and of the Results in Sect.3 Appendix 2: Proof of Theorem 4 References Conditional Distribution-Based Downweighting for Robust Estimation of Logistic Regression Models 1 Introduction 2 M-Estimators for Logistic Regression 2.1 A New Perspective to Outlier Downweighting 3 Modified Mallows Class Approach 4 Numerical Study 4.1 Simulation Settings 4.2 Simulation Results 4.3 Leukemia Dataset 5 Discussion References Bias Calibration for Robust Estimation in Small Areas 1 Introduction 2 General Framework and Notation 3 Bias Calibration for Non-linear Parameter Estimates 3.1 Linearization by the Influence Function 4 Model-based Simulation Study 5 Some Practical Issues 5.1 Full Calibration vs. Partial Calibration 5.2 Choice of the Tuning Parameters 6 Estimating the Gini Coefficient for Labor Market Areas in Tuscany 6.1 Results for LMAs in Sample Areas with Partial Calibration 6.2 Results for All Areas with Full Calibration 7 Conclusions and Further Discussion Appendix Influence Function of the Gini Coefficient Bootstrap for RMSE and Tuning Parameter Selection Details on the Estimator for Tuning Parameters References The Diverging Definition of Robustness in Statistics and Computer Vision 1 Collaborations 2 Statistics vs. Computer Vision 3 RANSAC 4 MISRE 5 Possible Cooperation References Part IV Other Methods Power Calculations and Critical Values for Two-Stage Nonparametric Testing Regimes 1 Introduction 2 Assumptions 3 Existing Probability Calculations 4 Approximating Corner Probabilities 5 Existing Approximate Critical Values 6 A New Bivariate Quantile Approximation 7 Application to Rank Tests 8 Continuity Correction for the Two-Stage Wilcoxon Statistic 9 Sample Size Calculation 10 An Example Calculation 11 Results 12 Errors in Levels for Approximate Critical Values 13 Errors in Approximations to Power 14 Conclusions Appendix 1: A Bivariate Recursion for Exact Probabilities Appendix 2: A Continuous Example with Nonzero Skewness References Data Nuggets in Supervised Learning 1 Introduction 1.1 Literature Overview 1.2 Data Nuggets 2 Setup 2.1 Formation of Nuggets 2.2 Estimation with Nuggets 3 Asymptotics 3.1 Intuition 3.2 Consistency of Coefficient Estimator 3.3 Consistency of Variance Estimator 3.4 Asymptotic Normality of Coefficient Estimator 4 Example 5 Simulations 5.1 First Simulation Set: Prediction 5.2 Second Simulation Set: Estimation 6 Extensions 7 Discussion 8 Conclusion References Improved Convergence Rates of Normal Extremes 1 Introduction 2 Main Results 2.1 Pointwise Convergence Rates 2.2 Uniform Convergence Rate 2.3 Comparisons of Different Convergence Rates 2.4 k-th Maxima 3 Applications and Numerical Comparisons 3.1 Numerical Comparisons 3.2 An Example 4 Conclusion Appendix Expansion of bn Additional Figures References Local Spectral Analysis of Qualitative Sequences via Minimum Description Length 1 Introduction 2 Spectral Envelope 2.1 Estimation 2.2 An Example 3 Local Analysis 3.1 Local Whittle Likelihood 3.2 Minimum Description Length 3.3 Optimization via Genetic Algorithm 3.4 Another Example Data Availability References