Statistical analysis of observations of increasing dimension. Transl. from the Russian
[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 literature. For instance, ``large numbers law'' should have been ``law of large numbers'', ``invariant principle'' should have been ``invariance principle'', ``Distribution density'' should have been ``probability density'', ``Two multiply eigenvalues'' should have been ``eigenvalues of multiplicity two'', etc. There are too many to be listed here. The publisher should have taken pains to employ a good language editor familiar with technical translation of mathematics in Russian to mathematics in English. The book has innovative technical ideas useful for those working in multivariate statistical analysis. It is too expensive for individuals to buy it.
perturbation theory for linear operators
scientific article; zbMATH DE number 920072