Large Covariance and Autocovariance Matrices (Chapman & Hall/CRC Monographs on Statistics and Applied Probability)

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. Cover Title Preface Introduction Part I Chapterr1 LARGE COVARIANCE MATRIX I 1.1 Consistency 1.2 Covariance classes and regularization 1.2.1 Covariance classes 1.2.2 Covariance regularization 1.3 Bandable p 1.3.1 Parameter space 1.3.2 Estimation in U 1.3.3 Minimaxity 1.4 Toeplitz p 1.4.1 Parameter space 1.4.2 Estimation in Gf(M) or Ff(M0;M) 1.4.3 Minimaxity 1.5 Sparse p 1.5.1 Parameter space 1.5.2 Estimation in U (q;C0(p);M) or Gq(Cn;p) 1.5.3 Minimaxity Chapterr2 LARGE COVARIANCE MATRIX II 2.1 Bandable p 2.1.1 Models and examples 2.1.2 Weak dependence 2.1.3 Estimation 2.2 Sparse p Chapterr3 LARGE AUTOCOVARIANCE MATRIX 3.1 Models and examples 3.2 Estimation of 􀀀0;p 3.3 Estimation of 􀀀u;p 3.3.1 Parameter spaces 3.3.2 Estimation 3.4 Estimation in MA(r) 3.5 Estimation in IVAR(r) 3.6 Gaussian assumption 3.7 Simulations Part II Chapterr4 SPECTRAL DISTRIBUTION 4.1 LSD 4.1.1 Moment method 4.1.2 Method of Stieltjes transform 4.2 Wigner matrix: Semi-circle law 4.3 Independent matrix: Marcenko{Pastur law 4.3.1 Results on Z: p=n ! y > 0 4.3.2 Results on Z: p=n ! 0 Chapterr5 NON-COMMUTATIVE PROBABILITY 5.1 NCP and its convergence 5.2 Essentials of partition theory 5.2.1 MŁobius function 5.2.2 Partition and non-crossing partition 5.2.3 Kreweras complement 5.3 Free cumulant; free independence 5.4 Moments of free variables 5.5 Joint convergence of random matrices 5.5.1 Compound free Poisson Chapterr6 GENERALIZED COVARIANCE MATRIX I 6.1 Preliminaries 6.1.1 Assumptions 6.1.2 Embedding 6.2 NCP convergence 6.2.1 Main idea 6.2.2 Main convergence 6.3 LSD of symmetric polynomials 6.4 Stieltjes transform 6.5 Corollaries Chapterr7 GENERALIZED COVARIANCE MATRIX II 7.1 Preliminaries 7.1.1 Assumptions 7.1.2 Centering and Scaling 7.1.3 Main idea 7.2 NCP convergence 7.3 LSD of symmetric polynomials 7.4 Stieltjes transform 7.5 Corollaries Part III Chapterr8 SPECTRA OF AUTOCOVARIANCE MATRIX I 8.1 Assumptions 8.2 LSD when p=n ! y 2 (0;1) 8.2.1 MA(q), q < 1 8.2.2 MA(1) 8.2.3 Application to specifc cases 8.3 LSD when p=n ! 0 8.3.1 Application to specifc cases 8.4 Non-symmetric polynomials Chapterr9 SPECTRA OF AUTOCOVARIANCE MATRIX II 9.1 Assumptions 9.2 LSD when p=n ! y 2 (0;1) 9.2.1 MA(q), q < 1 9.2.2 MA(1) 9.3 LSD when p=n ! 0 9.3.1 MA(q); q < 1 9.3.2 MA(1) Chapterr10 GRAPHICAL INFERENCE 10.1 MA order determination 10.2 AR order determination 10.3 Graphical tests for parameter matrices Chapterr11 TESTING WITH TRACE 11.1 One sample trace 11.2 Two sample trace 11.3 Testing Appendix: SUPPLEMENTARY PROOFS A.1 Proof of Lemma 6.3.1 A.2 Proof of Theorem 6.4.1(a) A.3 Proof of Theorem 7.2 A.4 Proof of Lemma 8.2.1 A.5 Proof of Corollary 8.2.1(c) A.6 Proof of Corollary 8.2.4(c) A.7 Proof of Corollary 8.3.1(c) A.8 Proof of Lemma 8.2.2 A.9 Proof of Lemma 8.2.3 A.10 Lemmas for Theorem 8.2.2 Bibliography Index 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. -- Provided by publisher