Large Covariance and Autocovariance Matrices

ISBN-13: 9780367734107 / Angielski / Miękka / 2020 / 272 str.

Arup Bose; Monika Bhattacharjee

Opis

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 avai

Spis treści

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

O autorze

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.