The material will be based on very recent advances in the theory and application of large covariance and autocovariance matrices. Technologies and methods in medical sciences, image processing, and other fields generate data where the dimension is large compared to the sample size and may also increase as the next set of measurements become available. Theoretical and practical study of such type of data has attracted recent attention of researchers since most of the methods in finite dimensional set up do not work in these cases, even asymptotically. This book will be mainly focused on the topics in high-dimensional situations.…mehr
The material will be based on very recent advances in the theory and application of large covariance and autocovariance matrices. Technologies and methods in medical sciences, image processing, and other fields generate data where the dimension is large compared to the sample size and may also increase as the next set of measurements become available. Theoretical and practical study of such type of data has attracted recent attention of researchers since most of the methods in finite dimensional set up do not work in these cases, even asymptotically. This book will be mainly focused on the topics in high-dimensional situations.
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 Sankhy¿ 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.
Inhaltsangabe
1. LARGE COVARIANCE MATRIX I Consistency Covariance classes and regularization Covariance classes Covariance regularization Bandable p Parameter space Estimation in U Minimaxity Toeplitz p Parameter space Estimation in Gß (M ) or Fß (M0, M ) Minimaxity Sparse p Parameter space Estimation in U (q, C0(p), M ) or Gq (Cn,p) Minimaxity 2. LARGE COVARIANCE MATRIX II Bandable p Models and examples Weak dependence Estimation Sparse p 3. LARGE AUTOCOVARIANCE MATRIX Models and examples Estimation of 0,p Estimation of u,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: Mar cenko-Pastur law > 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
1. LARGE COVARIANCE MATRIX I Consistency Covariance classes and regularization Covariance classes Covariance regularization Bandable p Parameter space Estimation in U Minimaxity Toeplitz p Parameter space Estimation in Gß (M ) or Fß (M0, M ) Minimaxity Sparse p Parameter space Estimation in U (q, C0(p), M ) or Gq (Cn,p) Minimaxity 2. LARGE COVARIANCE MATRIX II Bandable p Models and examples Weak dependence Estimation Sparse p 3. LARGE AUTOCOVARIANCE MATRIX Models and examples Estimation of 0,p Estimation of u,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: Mar cenko-Pastur law > 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
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