Statistical analysis of observations of increasing dimension. Transl. from the Russian
type: Item
instance of: scholarly article
title: Statistical analysis of observations of increasing dimension. Transl. from the Russian
zbMATH Open document ID: 0867.62047
published in: Theory and Decision Library. Series B: Mathematical and Statistical Methods
publication date: 1996-08-26T00:00:00Z
review text: [For the review of the Russian original from 1988, see Zbl 0699.62052.] Consider the following statistical model: \(X_{n1},\dots, X_{nn}\) are i.i.d. \(m_n\)-dimensional random vectors with covariance matrix \(R_{m_n}\) where the sequences \(\{m_n\}\) and \(\{n\}\) satisfy the \(G\)-condition: \(\lim_{n\to\infty} f(m_n,n)<\infty\), and \(f(x,y)\) is a positive function increasing with respect to \(x\) and decreasing with respect to \(y\). For instance \(f(x,y)\) can be chosen to be \(xy^{-1}\). In other words the dimension \(m_n\) of the random vectors can increase with the growth of the number \(n\) of observations subject to the \(G\)-condition. It is further assumed that the dimension of the parameter space involved in the model remains constant with respect to \(n\) even when the dimension \(m_n\) of the random vectors \(\{X_{ni}\}\) increases. Under this setup, the author studies statistical analysis of observations. The contents are as follows: Chapter 1: Introduction to general statistical analysis; Chapter 2: The canonical equations \(C_1,\dots, C_3\) for the empirical covariance matrix; Chapter 4: Limit theorems for the eigenvalues of empirical covariance matrices; Chapter 5: \(G_2\)-estimator for the Stieltjes transform of the normalized spectral function of covariance matrices; Chapter 6: Statistical estimators for solutions of systems of linear algebraic equations. The problems studied are interesting and they have not been dealt with in any other book as far as the reviewer is aware. The author has brought in techniques from ``Perturbation theory for linear operators'', ``The theory of eigenvalues of random matrices'' and ``The theory of random determinants''. The book is very poorly translated by the author from his own Russian version of the book and apparently the publishers also did not employ a good language editor. The terminology at several places does not match with the standard terminology used in similar books available in the English language…
Mathematics Subject Classification ID: 62H12
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 47N30
zbMATH DE Number: 920072
zbMATH Keywords: G-condition
zbMATH Keywords: perturbation theory for linear operators
zbMATH Keywords: canonical equations
zbMATH Keywords: eigenvalues
zbMATH Keywords: empirical covariance matrices
zbMATH Keywords: Stieltjes transform
zbMATH Keywords: spectral function
zbMATH Keywords: random determinants
author: Vyacheslav Girko
reviewed by: B. L. S. Prakasa Rao
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 920072