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.
