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Full Description
Large Covariance and Autocovariance Matrices brings together a collection of recent results on sample covariance and autocovariance matrices in high-dimensional models and novel ideas on how to use them for statistical inference in one or more high-dimensional time series models. The prerequisites include knowledge of elementary multivariate analysis, basic time series analysis and basic results in stochastic convergence.
Part I is on different methods of estimation of large covariance matrices and auto-covariance matrices and properties of these estimators. Part II covers the relevant material on random matrix theory and non-commutative probability. Part III provides results on limit spectra and asymptotic normality of traces of symmetric matrix polynomial functions of sample auto-covariance matrices in high-dimensional linear time series models. These are used to develop graphical and significance tests for different hypotheses involving one or more independent high-dimensional linear time series.
The book should be of interest to people in econometrics and statistics (large covariance matrices and high-dimensional time series), mathematics (random matrices and free probability) and computer science (wireless communication). Parts of it can be used in post-graduate courses on high-dimensional statistical inference, high-dimensional random matrices and high-dimensional time series models. It should be particularly attractive to researchers developing statistical methods in high-dimensional time series models.
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.
Contents
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 Theorem
Proof of Lemma
Proof of Corollary (c)
Proof of Corollary (c)
Proof of Corollary (c)
Proof of Lemma
Proof of Lemma
Lemmas for Theorem
