Chapter 1. Introduction: the Development of Multiparametric Statistics
The Stein effect
The Kolmogorov Asymptotics
Spectral Theory of Increasing Random Matrices
Constructing Multiparametric Procedures
Optimal Solution to Empirical Linear Equations

Chapter 2. Fundamental Problem of Statistics

2.1. Shrinkage of Sample Mean Vector

Shrinkage for Normal Distributions
Shrinkage for a Wide Class of Distributions

2.2. Shrinkage of Unbiased Estimators

Special Shrinkage of Normal Estimators
Shrinkage of Arbitrary Unbiased Estimators
Limit Quadratic Risk of Shrinkage Estimators

2.3. Shrinkage of Infinite-Dimensional Vectors

Normal distributions
Wide Class of Distributions

2.4. Unimprovable Component-Wise Estimation

Estimator for the Density of Parameters
Estimator for the Best Estimating Function

Chapter 3. Spectral Theory of Large Sample Covariance Matrices

3.1. Spectral Properties of Large Sample Covariance Matrices\page{64}

Gram Matrices
Sample Covariance Matrices
Limit Spectra

3.2. Spectral Functions of Infinite Sample Covariance Matrices

Dispersion Equations for Infinite Gram Matrices
Dispersion Equations for Sample Covariance Matrices
Limit Spectral Equations

3.3. Normalization of Quality Functions

Spectral Functions of Sample Covariance Matrices
Normal Evaluation of Sample Dependent Functionals

Chapter 4. Asymptotically Unimprovable Solution of Multivariate Problems

4.1. Estimators of Large Inverse Covariance Matrices

Problem Setting
Shrinkage for Inverse Covariance Matrices
Generalized Ridge-Estimators
Asymptotically Unimprovable Estimator
Proofs for Section 4.1

4.2. Matrix Shrinkage Estimators of Expectation Vectors

Limit Quadratic Risk for Estimators of Vectors
Minimization of the Limit Quadratic Risk
Statistics to Approximate Limit Risk
Statistics to Approximate the Extremum Solution\page{138}

4.3. Multiparametric Sample Linear Regression

Some Spectral Functions of Sample Covariance Matrices
Functionals of Random Gram Matrices
Functionals in the Regression Problem
Minimization of Quadratic Risk
Special Cases

Chapter 5. Multiparametric Discriminant Analysis \rm

5.1. Discriminant Analysis of Independent Variables

A priori Weighting of Variables
Empirical Weighting of Variables
Error Probability for Empirical Weighting
Statistics to Estimate Probabilities of Errors
Contribution of a Small Number of Variables
Selection of Variables by Threshold

5.2. Discriminant Analysis of Dependent Variables

Asymptotical Setting
Moments of Generalized Discriminant Function
Limit Probabilities of Errors
Best-in-the-Limit Discriminant Procedure
The Extension to a Wide Class of Distributions

Chapter 6. Theory of Solution to High-Order Systems
of Empirical Linear Algebraic Equations

6.1. The Best Bayes Solution\page{204}
6.2. Asymptotically Unimprovable Solution

Spectral Functions of Large Gram Matrices
Limit Spectral Functions of Gram Matrices
Quadratic Risk of Pseudosolutions
Minimization of the Limit Risk
Shrinkage--Ridge Pseudosolution
Proofs for Section 6.2


Experimental Investigation of Spectral Functions
of Large Sample Covariance Matrices


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