In the last few decades the accumulation of large amounts of in­ formation in numerous applications. has stimtllated an increased in­ terest in multivariate analysis. Computer technologies allow one to use multi-dimensional and multi-parametric models successfully. At the same time, an interest arose in statistical analysis with a de­ ficiency of sample data. Nevertheless, it is difficult to describe the recent state of affairs in applied multivariate methods as satisfactory. Unimprovable (dominating) statistical procedures are still unknown except for a few specific cases. The simplest problem of estimat­ ing the mean vector with minimum quadratic risk is unsolved, even for normal distributions. Commonly used standard linear multivari­ ate procedures based on the inversion of sample covariance matrices can lead to unstable results or provide no solution in dependence of data. Programs included in standard statistical packages cannot process 'multi-collinear data' and there are no theoretical recommen­ dations except to ignore a part of the data. The probability of data degeneration increases with the dimension n, and for n > N, where N is the sample size, the sample covariance matrix has no inverse. Thus nearly all conventional linear methods of multivariate statis­ tics prove to be unreliable or even not applicable to high-dimensional data.

Table of Contents:
Kolmogorov Asymptotics in Problems of Multivariate Analysis.- Spectral Theory of Large Covariance Matrices.- Approximately Unimprovable Essentially Multivariate Procedures.- 1. Spectral Properties of Large Wishart Matrices.- Wishart Distribution.- Limit Moments of Wishart Matrices.- Limit Formula for the Resolvent of Wishart Matrices.- 2. Resolvents and Spectral Functions of Large Sample Covariance Matrices.- Spectral Functions of Random Gram Matrices.- Spectral Functions of Sample Covariance Matrices.- Limit Spectral Functions of the Increasing Sample Covariance Matrices.- 3. Resolvents and Spectral Functions of Large Pooled Sample Covariance Matrices.- Problem Setting.- Spectral Functions of Pooled Random Gram Matrices.- Spectral Functions of Pooled Sample Covariance Matrices.- Limit Spectral Functions of the Increasing Pooled Sample Covariance Matrices.- 4. Normal Evaluation of Quality Functions.- Measure of Normalizability.- Spectral Functions of Large Covariance Matrices.- Normal Evaluation of Sample Dependent Functionals.- Discussion.- 5. Estimation of High-Dimensional Inverse Covariance Matrices.- Shrinkage Estimators of the Inverse Covariance Matrices.- Generalized Ridge Estimators of the Inverse Covariance Matrices.- Asymptotically Unimprovable Estimators of the Inverse Covariance Matrices.- 6. Epsilon-Dominating Component-Wise Shrinkage Estimators of Normal Mean.- Estimation Function for the Component-Wise Estimators.- Estimators of the Unimprovable Estimation Function.- 7. Improved Estimators of High-Dimensional Expectation Vectors.- Limit Quadratic Risk for a Class of Estimators of Expectation Vectors.- Minimization of the Limit Quadratic Risk.- Statistics to Approximate the Limit Risk Function.- Statistics to Approximate the Extremal limit Solution.- 8. Quadratic Risk of Linear Regression with a Large Number of Random Predictors.- Spectral Functions of Sample Covariance Matrices.- Functionals Depending on the Statistics Sand ?0.- Functionals Depending on Sample Covariance Matrices and Covariance Vectors.- The Leading Part of the Quadratic Risk and its Estimator.- Special Cases.- 9. Linear Discriminant Analysis of Normal Populations with Coinciding Covariance Matrices.- Problem Setting.- Expectation and Variance of Generalized Discriminant Functions.- Limit Probabilities of the Discrimination Errors.- 10. Population Free Quality of Discrimination.- Problem Setting.- Leading Parts of Functionals for Normal Populations.- Leading Parts of Functionals for Arbitrary Populations.- Discussion.- Proofs.- 11. Theory of Discriminant Analysis of the Increasing Number of Independent Variables.- Problem Setting.- A Priori Weighting of Independent Variables.- Minimization of the Limit Error Probability for a Priori Weighting.- Weighting of Independent Variables by Estimators.- Minimization of the Limit Error Probability for Weighting by Estimators.- Statistics to Estimate Probabilities of Errors.- Contribution of Variables to Discrimination.- Selection of a Large Number of Independent Variables.- Conclusions.- References.