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Multivariate Statistical Analysis : A High-Dimensional Approach - V.I. Serdobolskii

Multivariate Statistical Analysis

A High-Dimensional Approach


Published: 31st October 2000
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Presents a new branch of mathematical statistics aimed at constructing unimprovable methods of multivariate analysis, multi-parametric estimation, and discriminant and regression analysis. In contrast to the traditional consistent Fisher method of statistics, the essentially multivariate technique is based on the decision function approach by A. Wald. Developing this new method for high dimensions, comparable in magnitude with sample size, provides stable approximately unimprovable procedures in some wide classes, depending on an arbitrary function. A fact is established that, for high-dimensional problems, under some weak restrictions on the variable dependence, the standard quality functions of regularized multivariate procedures prove to be independent of distributions. This opens the possibility to construct unimprovable procedures free from distributions.

Prefacep. ix
Introductionp. 1
Kolmogorov Asymptotics in Problems of Multivariate Analysisp. 2
Spectral Theory of Large Covariance Matricesp. 5
Limit Formulas for Spectral Functionsp. 5
Spectral Functions for Fixed Dimension and Sample Sizep. 11
Method to Single out the Leading Termsp. 13
Approximately Unimprovable Essentially Multivariate Proceduresp. 14
Spectral Properties of Large Wishart Matricesp. 25
Wishart Distributionp. 27
Limit Moments of Wishart Matricesp. 32
Limit Formula for the Resolvent of Wishart Matricesp. 38
Resolvents and Spectral Functions of Large Sample Covariance Matricesp. 40
Spectral Functions of Random Gram Matricesp. 41
Spectral Functions of Sample Covariance Matricesp. 47
Limit Spectral Functions of the Increasing Sample Covariance Matricesp. 52
Resolvents and Spectral Functions of Large Pooled Sample Covariance Matricesp. 61
Problem Settingp. 61
Spectral Functions of Pooled Random Gram Matricesp. 63
Spectral Functions of Pooled Sample Covariance Matricesp. 68
Limit Spectral Functions of the Increasing Pooled Sample Covariance Matricesp. 73
Normal Evaluation of Quality Functionsp. 76
Measure of Normalizabilityp. 77
Spectral Functions of Large Covariance Matricesp. 78
Normal Evaluation of Sample Dependent Functionalsp. 79
Discussionp. 86
Estimation of High-Dimensional Inverse Covariance Matricesp. 87
Shrinkage Estimators of the Inverse Covariance Matricesp. 88
Generalized Ridge Estimators of the Inverse Covariance Matricesp. 89
Asymptotically Unimprovable Estimators of the Inverse Covariance Matricesp. 98
Epsilon-Dominating Component-Wise Shrinkage Estimators of Normal Meanp. 102
Estimation Function for the Component-Wise Estimatorsp. 102
Estimators of the Unimprovable Estimation Functionp. 104
Improved Estimators of High-Dimensional Expectation Vectorsp. 112
Limit Quadratic Risk for a Class of Estimators of Expectation Vectorsp. 113
Minimization of the Limit Quadratic Riskp. 119
Statistics to Approximate the Limit Risk Functionp. 124
Statistics to Approximate the Extremal limit Solutionp. 126
Quadratic Risk of Linear Regression with a Large Number of Random Predictorsp. 131
Spectral Functions of Sample Covariance Matricesp. 133
Functionals Depending on the Statistics S and g0p. 135
Functionals Depending on Sample Covariance Matrices and Covariance Vectorsp. 144
The Leading Part of the Quadratic Risk and its Estimatorp. 148
Special Casesp. 153
Linear Discriminant Analysis of Normal Populations with Coinciding Covariance Matricesp. 156
Problem Settingp. 157
Expectation and Variance of Generalized Discriminant Functionsp. 159
Limit Probabilities of the Discrimination Errorsp. 166
Population Free Quality of Discriminationp. 169
Problem Settingp. 169
Leading Parts of Functionals for Normal Populationsp. 171
Leading Parts of Functionals for Arbitrary Populationsp. 173
Discussionp. 176
Proofsp. 177
Theory of Discriminant Analysis of the Increasing Number of Independent Variablesp. 187
Problem Settingp. 187
A Priori Weighting of Independent Variablesp. 193
Minimization of the Limit Error Probability for a Priori Weightingp. 201
Weighting of Independent Variables by Estimatorsp. 203
Minimization of the Limit Error Probability for Weighting by Estimatorsp. 210
Statistics to Estimate Probabilities of Errorsp. 214
Contribution of Variables to Discriminationp. 217
Selection of a Large Number of Independent Variablesp. 219
Conclusionsp. 227
Referencesp. 233
Indexp. 239
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780792366430
ISBN-10: 0792366433
Series: Theory and Decision Library B
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 244
Published: 31st October 2000
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.4 x 15.6  x 1.91
Weight (kg): 1.2