High-Dimensional Covariance Matrix Estimation: An Introduction to Random Matrix Theory

Description

This book presents covariance matrix estimation and related aspects of random matrix theory. It focuses on the sample covariance matrix estimator and provides a holistic description of its properties under two asymptotic regimes: the traditional one, and the high-dimensional regime that better fits the big data context. It draws attention to the deficiencies of standard statistical tools when used in the high-dimensional setting, and introduces the basic concepts and major results related to spectral statistics and random matrix theory under high-dimensional asymptotics in an understandable and reader-friendly way. The aim of this book is to inspire applied statisticians, econometricians, and machine learning practitioners who analyze high-dimensional data to apply the recent developments in their work. Foreword Acknowledgments General Conventions and Notation Contents 1 Introduction References 2 Traditional Estimators and Standard Asymptotics 2.1 Sample Covariance Matrix 2.2 Sample Eigenvalues 2.2.1 Exact Distribution 2.2.2 Finite Sample Bias 2.2.3 Asymptotic Distribution 2.3 Sample Eigenvectors 2.4 Sphericity and Partial Sphericity Tests References 3 Finite Sample Performance of Traditional Estimators 3.1 Sample Covariance Matrix 3.2 Sample Eigenvalues 3.3 Sample Eigenvectors 3.4 PCA Applications 3.5 Sphericity Test 3.6 Partial Sphericity Test References 4 Traditional Estimators and High-Dimensional Asymptotics 4.1 Sample Covariance Matrix 4.2 Sample Eigenvalues 4.2.1 Bounds on λn,1 and λn,min 4.2.2 Fluctuation Results for λn,1 and λn,min 4.2.3 Marčcenko and Pastur (1967) Result 4.2.4 Spiked Population Model by Johnstone (2001) 4.2.5 Generalized Spiked Population Model of Bai and Yao (2012) 4.3 Sample Eigenvectors 4.4 PCA Applications 4.5 Centered Sample Covariance Matrix and Substitution Principle 4.6 Sphericity Test 4.7 Partial Sphericity Test 4.7.1 Kritchman and Nadler (2009) Test 4.7.2 Passemier and Yao (2014) Test References 5 Summary and Outlook Comparative Study of Statistical Properties Under Two Asymptotic Regimes. Traditional Estimators and LRT Statistics References A Graphs B Tables C Additional Theoretical Results C.1 Spiked Population Model C.2 Generalized Spiked Population Model: Silverstein Equation C.3 Generalized Spiked Population Model: Multiplicities of Spikes mi(τi) > 1, i = 1, …, k C.4 Partial Sphericity Test C.5 Passemier and Yao Test Calibration References