Ebook: 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 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.

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This book 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 remarkable fact is established: 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. For the first time in the history of statistics, this opens the possibility to construct unimprovable procedures free from distributions.
Audience: This work will be of interest to researchers and graduate students whose work involves statistics and probability, reliability and risk analysis, econometrics, machine learning, medical statistics, and various applications of multivariate analysis.

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This book 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 remarkable fact is established: 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. For the first time in the history of statistics, this opens the possibility to construct unimprovable procedures free from distributions.
Audience: This work will be of interest to researchers and graduate students whose work involves statistics and probability, reliability and risk analysis, econometrics, machine learning, medical statistics, and various applications of multivariate analysis.
Content:
Front Matter....Pages i-xii
Introduction....Pages 1-24
Spectral Properties of Large Wishart Matrices....Pages 25-39
Resolvents and Spectral Functions of Large Sample Covariance Matrices....Pages 40-60
Resolvent and Spectral Functions of Large Pooled Sample Covariance Matrices....Pages 61-75
Normal Evaluation of Quality Functions....Pages 76-86
Estimation of High-Dimensional Inverse Covariance Matrices....Pages 87-101
Epsilon-Dominating Component-Wise Shrinkage Estimators of Normal Mean....Pages 102-111
Improved Estimators of High-Dimensional Expectation Vectors....Pages 112-130
Quadratic Risk of Linear Regression with a Large Number of Random Predictors....Pages 131-155
Linear Discriminant Analysis of Normal Populations with Coinciding Covariance Matrices....Pages 156-168
Population Free Quality of Linear Discrimination....Pages 169-186
Theory of Discriminant Analysis of the Increasing Number of Independent Variables....Pages 187-226
Conclusions....Pages 227-232
Back Matter....Pages 233-244

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This book 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 remarkable fact is established: 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. For the first time in the history of statistics, this opens the possibility to construct unimprovable procedures free from distributions.
Audience: This work will be of interest to researchers and graduate students whose work involves statistics and probability, reliability and risk analysis, econometrics, machine learning, medical statistics, and various applications of multivariate analysis.
Content:
Front Matter....Pages i-xii
Introduction....Pages 1-24
Spectral Properties of Large Wishart Matrices....Pages 25-39
Resolvents and Spectral Functions of Large Sample Covariance Matrices....Pages 40-60
Resolvent and Spectral Functions of Large Pooled Sample Covariance Matrices....Pages 61-75
Normal Evaluation of Quality Functions....Pages 76-86
Estimation of High-Dimensional Inverse Covariance Matrices....Pages 87-101
Epsilon-Dominating Component-Wise Shrinkage Estimators of Normal Mean....Pages 102-111
Improved Estimators of High-Dimensional Expectation Vectors....Pages 112-130
Quadratic Risk of Linear Regression with a Large Number of Random Predictors....Pages 131-155
Linear Discriminant Analysis of Normal Populations with Coinciding Covariance Matrices....Pages 156-168
Population Free Quality of Linear Discrimination....Pages 169-186
Theory of Discriminant Analysis of the Increasing Number of Independent Variables....Pages 187-226
Conclusions....Pages 227-232
Back Matter....Pages 233-244
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