1st Edition
Large Covariance and Autocovariance Matrices
1. LARGE COVARIANCE MATRIX I
Consistency
Covariance classes and regularization
Covariance classes
Covariance regularization
Bandable Σp
Parameter space
Estimation in U
Minimaxity
Toeplitz Σp
Parameter space
Estimation in Gβ (M ) or Fβ (M0, M )
Minimaxity
Sparse Σp
Parameter space
Estimation in Uτ (q, C0(p), M ) or Gq (Cn,p)
Minimaxity
2. LARGE COVARIANCE MATRIX II
Bandable Σp
Models and examples
Weak dependence
Estimation
Sparse Σp
3. LARGE AUTOCOVARIANCE MATRIX
Models and examples
Estimation of Γ0,p
Estimation of Γu,p
Parameter spaces
Estimation
Estimation in MA(r)
Estimation in IVAR(r)
Gaussian assumption
Simulations
Part II
4. SPECTRAL DISTRIBUTION
LSD
Moment method
Method of Stieltjes transform
Wigner matrix: semi-circle law
Independent matrix: Marˇcenko-Pastur law
Results on Z: p/n → y > 0
Results on Z: p/n → 0
5. NON-COMMUTATIVE PROBABILITY
NCP and its convergence
Essentials of partition theory
M¨obius function
Partition and non-crossing partition
Kreweras complement
Free cumulant; free independence
Moments of free variables
Joint convergence of random matrices
Compound free Poisson
6. GENERALIZED COVARIANCE MATRIX I
Preliminaries
Assumptions
Embedding
NCP convergence
Main idea
Main convergence
LSD of symmetric polynomials
Stieltjes transform
Corollaries
7. GENERALIZED COVARIANCE MATRIX II
Preliminaries
Assumptions
Centering and Scaling
Main idea
NCP convergence
LSD of symmetric polynomials
Stieltjes transform
Corollaries
8. SPECTRA OF AUTOCOVARIANCE MATRIX I
Assumptions
LSD when p/n → y ∈ (0, ∞)
MA(q), q < ∞
MA(∞)
Application to specific cases
LSD when p/n → 0
Application to specific cases
Non-symmetric polynomials
9. SPECTRA OF AUTOCOVARIANCE MATRIX II
Assumptions
LSD when p/n → y ∈ (0, ∞)
MA(q), q < ∞
MA(∞)
LSD when p/n → 0
MA(q), q < ∞
MA(∞)
10. GRAPHICAL INFERENCE
MA order determination
AR order determination
Graphical tests for parameter matrices
11. TESTING WITH TRACE
One sample trace
Two sample trace
Testing
12. SUPPLEMENTARY PROOFS
Proof of Lemma
Proof of Theorem (a)
Proof of Th
Biography
Arup Bose is a professor at the Indian Statistical Institute, Kolkata, India. He is a distinguished researcher in mathematical statistics and has been working in high-dimensional random matrices for the last fifteen years. He has been editor of Sankhyā for several years and has been on the editorial board of several other journals. He is a Fellow of the Institute of Mathematical Statistics, USA and all three national science academies of India, as well as the recipient of the S.S. Bhatnagar Award and the C.R. Rao Award. His first book Patterned Random Matrices was also published by Chapman & Hall. He has a forthcoming graduate text U-statistics, M-estimates and Resampling (with Snigdhansu Chatterjee) to be published by Hindustan Book Agency.
Monika Bhattacharjee is a post-doctoral fellow at the Informatics Institute, University of Florida. After graduating from St. Xavier's College, Kolkata, she obtained her master’s in 2012 and PhD in 2016 from the Indian Statistical Institute. Her thesis in high-dimensional covariance and auto-covariance matrices, written under the supervision of Dr. Bose, has received high acclaim.
" . . . the authors should be congratulated for producing two highly relevant and well-written books. Statisticians would probably gravitate to LCAM in the first instance and those working in linear algebra would probably gravitate to PRM."
~Jonathan Gillard, Cardiff University"The book represents a monograph of the authors’ recent results about the theory of large covariance and autocovariance matrices and contains other important results from other research papers and books in this topic. It is very useful for all researchers who use large covariance and autocovariance matrices in their researches. Especially, it is very useful for post-graduate and PhD students in mathematics, statistics, econometrics and computer science. It is a well-written and organized book with a large number of solved examples and many exercises left to readers for homework. I would like to recommend the book to PhD students and researchers who want to learn or use large covariance and autocovariance matrices in their researches."
~ Miroslav M. Ristic (Niš), zbMath"This book brings together a collection of recent results on estimation of multidimensional time series covariance matrices. In the case where the time series consists of a sequence of independent (Chapter 1) or weakly dependent (Chapter 2) random vectors, the authors call it covariance estimation, whereas in the general case where the time series is only stationary, they call it autocovariance estimation. The framework of the results presented here is the one where the dimension of the observations (as well as the observation window size, otherwise nothing can be said) is high. The prerequisites include knowledge of elementary multivariate analysis, basic time series analysis, and basic results in stochastic convergence.
In Chapter 1, the authors consider the case where we have at our disposal a large time series of iid high-dimensional observations with common covariance
