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Multivariate Statistical Analysis

A High-Dimensional Approach

  • Book
  • © 2000

Overview

Authors:
  1. V. Serdobolskii

    1. Department of Applied Mathematics, Moscow Institute of Electronics and Mathematics, Moscow, Russia

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Part of the book series: Theory and Decision Library B (TDLB, volume 41)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • V. Serdobolskii
    Pages 1-24

  3. Spectral Properties of Large Wishart Matrices

    • V. Serdobolskii
    Pages 25-39

  4. Resolvents and Spectral Functions of Large Sample Covariance Matrices

    • V. Serdobolskii
    Pages 40-60

  5. Resolvent and Spectral Functions of Large Pooled Sample Covariance Matrices

    • V. Serdobolskii
    Pages 61-75

  6. Normal Evaluation of Quality Functions

    • V. Serdobolskii
    Pages 76-86

  7. Estimation of High-Dimensional Inverse Covariance Matrices

    • V. Serdobolskii
    Pages 87-101

  8. Epsilon-Dominating Component-Wise Shrinkage Estimators of Normal Mean

    • V. Serdobolskii
    Pages 102-111

  9. Improved Estimators of High-Dimensional Expectation Vectors

    • V. Serdobolskii
    Pages 112-130

  10. Quadratic Risk of Linear Regression with a Large Number of Random Predictors

    • V. Serdobolskii
    Pages 131-155

  11. Linear Discriminant Analysis of Normal Populations with Coinciding Covariance Matrices

    • V. Serdobolskii
    Pages 156-168

  12. Population Free Quality of Linear Discrimination

    • V. Serdobolskii
    Pages 169-186

  13. Theory of Discriminant Analysis of the Increasing Number of Independent Variables

    • V. Serdobolskii
    Pages 187-226

  14. Conclusions

    • V. Serdobolskii
    Pages 227-232

  15. Back Matter

    Pages 233-244
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Applied Mathematics, Moscow Institute of Electronics and Mathematics, Moscow, Russia

    V. Serdobolskii

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