Multivariate Statistical Analysis A High-Dimensional Approach

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 aros...

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Bibliographic Details
Main Author: Serdobolskii, V.I.
Format: eBook
Language:English
Published: Dordrecht Springer Netherlands 2000, 2000
Edition:1st ed. 2000
Series:Theory and Decision Library B, Mathematical and Statistical Methods
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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 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