The investigation of the leading terms of the increasing dimension asymptotics led to the construction of a systematic theory of multivariate analysis characterized by other settings, specific problems, and results of interest for applications. A statistical problem in which the dimension of observations is comparable to the sample size may be called an \it essentially multivariate problem. \rm The statistical analysis taking into account finite effects produced by the estimation of a large number of parameters and related to the solution of essentially multivariate problems may be called the \it essentially multivariate analysis.

The central idea of the investigation of essentially multivariate effects is to study relations between empirical distribution functions of true parameters and of their estimators. Limit equations are derived that connect spectral functions of sample covariance matrices and of true covariance matrices. Such relations proved to be of a special interest for the essentially multivariate approach, since they present a device for a regular construction of improved estimators in different multivariate problems. Using these relations one can first single out non-random leading parts of quality functionals involved in multivariate analysis and then construct their consistent estimators. To obtain an improved procedure, it suffices to maximize these estimators.

The book consists of an Introduction and twelve chapters. The introduction presents historical aspects and the line of development of main ideas. In Chapter 1 the reader will recall the fundamentals of the theory of multivariate analysis in the case when the underlying distributions are normal. In Chapters 2--11 the results of the original investigations are presented. These chapters are mostly independent of each other and written so that they can be read separately.

The author hopes that specialists in mathematical statistics will be interested in this new branch of the theory of statistics and in the new phenomena investigated. The essentially multivariate statistics is different in its approach, in its special techniques, and in its results of a new kind.

Applied statisticians and users of statistical software will be interested in more efficient methods of practical multivariate analysis that can be developed by using essentially multivariate methods. In fact, nearly all existing software for applied multivariate analysis is now obsolete. The essentially multivariate technique promises to provide stable, uniformly consistent with respect to the number of variables, approximately non-improvable methods whose quality does not depend on distributions.
Students of mathematics obtain a text-book, unique for today, for studying the recently created theory of more efficient methods of multivariate analysis. For a new generation of mathematicians this theory may undoubtedly serve as a reliable basis for their future success in the science of 21st century.

the author would like to express my sincere gratitude to Yurii Vasilievich Prokhorov for his attention, invariable support of my investigations, and wise recommendations. Also I am heartily thankful to Victor Matveevich Bukhshtaber for an enthusiastic attitude and a suggestion to write this book.

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