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Published: 31st October 2000
ISBN: 9780792366430
Number Of Pages: 244
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Published: 9th December 2010Presents 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.
| Preface | p. ix |
| Introduction | p. 1 |
| Kolmogorov Asymptotics in Problems of Multivariate Analysis | p. 2 |
| Spectral Theory of Large Covariance Matrices | p. 5 |
| Limit Formulas for Spectral Functions | p. 5 |
| Spectral Functions for Fixed Dimension and Sample Size | p. 11 |
| Method to Single out the Leading Terms | p. 13 |
| Approximately Unimprovable Essentially Multivariate Procedures | p. 14 |
| Spectral Properties of Large Wishart Matrices | p. 25 |
| Wishart Distribution | p. 27 |
| Limit Moments of Wishart Matrices | p. 32 |
| Limit Formula for the Resolvent of Wishart Matrices | p. 38 |
| Resolvents and Spectral Functions of Large Sample Covariance Matrices | p. 40 |
| Spectral Functions of Random Gram Matrices | p. 41 |
| Spectral Functions of Sample Covariance Matrices | p. 47 |
| Limit Spectral Functions of the Increasing Sample Covariance Matrices | p. 52 |
| Resolvents and Spectral Functions of Large Pooled Sample Covariance Matrices | p. 61 |
| Problem Setting | p. 61 |
| Spectral Functions of Pooled Random Gram Matrices | p. 63 |
| Spectral Functions of Pooled Sample Covariance Matrices | p. 68 |
| Limit Spectral Functions of the Increasing Pooled Sample Covariance Matrices | p. 73 |
| Normal Evaluation of Quality Functions | p. 76 |
| Measure of Normalizability | p. 77 |
| Spectral Functions of Large Covariance Matrices | p. 78 |
| Normal Evaluation of Sample Dependent Functionals | p. 79 |
| Discussion | p. 86 |
| Estimation of High-Dimensional Inverse Covariance Matrices | p. 87 |
| Shrinkage Estimators of the Inverse Covariance Matrices | p. 88 |
| Generalized Ridge Estimators of the Inverse Covariance Matrices | p. 89 |
| Asymptotically Unimprovable Estimators of the Inverse Covariance Matrices | p. 98 |
| Epsilon-Dominating Component-Wise Shrinkage Estimators of Normal Mean | p. 102 |
| Estimation Function for the Component-Wise Estimators | p. 102 |
| Estimators of the Unimprovable Estimation Function | p. 104 |
| Improved Estimators of High-Dimensional Expectation Vectors | p. 112 |
| Limit Quadratic Risk for a Class of Estimators of Expectation Vectors | p. 113 |
| Minimization of the Limit Quadratic Risk | p. 119 |
| Statistics to Approximate the Limit Risk Function | p. 124 |
| Statistics to Approximate the Extremal limit Solution | p. 126 |
| Quadratic Risk of Linear Regression with a Large Number of Random Predictors | p. 131 |
| Spectral Functions of Sample Covariance Matrices | p. 133 |
| Functionals Depending on the Statistics S and g0 | p. 135 |
| Functionals Depending on Sample Covariance Matrices and Covariance Vectors | p. 144 |
| The Leading Part of the Quadratic Risk and its Estimator | p. 148 |
| Special Cases | p. 153 |
| Linear Discriminant Analysis of Normal Populations with Coinciding Covariance Matrices | p. 156 |
| Problem Setting | p. 157 |
| Expectation and Variance of Generalized Discriminant Functions | p. 159 |
| Limit Probabilities of the Discrimination Errors | p. 166 |
| Population Free Quality of Discrimination | p. 169 |
| Problem Setting | p. 169 |
| Leading Parts of Functionals for Normal Populations | p. 171 |
| Leading Parts of Functionals for Arbitrary Populations | p. 173 |
| Discussion | p. 176 |
| Proofs | p. 177 |
| Theory of Discriminant Analysis of the Increasing Number of Independent Variables | p. 187 |
| Problem Setting | p. 187 |
| A Priori Weighting of Independent Variables | p. 193 |
| Minimization of the Limit Error Probability for a Priori Weighting | p. 201 |
| Weighting of Independent Variables by Estimators | p. 203 |
| Minimization of the Limit Error Probability for Weighting by Estimators | p. 210 |
| Statistics to Estimate Probabilities of Errors | p. 214 |
| Contribution of Variables to Discrimination | p. 217 |
| Selection of a Large Number of Independent Variables | p. 219 |
| Conclusions | p. 227 |
| References | p. 233 |
| Index | p. 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
