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V.Serdobolskii

MULTIPARAMETRIC STATISTICS

Announcement

The book presents mathematical theory of statistics using
models with essentially large number of
unknown parameters, i.e., so large that it is comparable with
sample size and can be much more.
In this meaning, the proposed theory can be called
"essentially multiparametric".
It starts a new branch of mathematical statistics
different by setting of problems, new phenomena, new methods of
investigation, and results of a new kind.
The main tool of the theory is the
Kolmogorov asymptotical approach in which
sample size increases along with the number of parameters and
their ratio tends to a constant. In this
situation, a number of specific phenomena are produced
including

--the appearance of stable relations between principal
parts of functions depending on a large number of random and
nonrandom arguments ("dispersion equations");

--a specific normalization effect
for quality functionals that are
proved to be the same as for normal distributions;

--the possibility of an evaluation
of quality functions from the observed data
making it possible to compare statistical procedures and choose better ones.

A regular technique is developed for implementing new more
efficient statistical methods: a class of non-degenerating
procedures is chosen, the limit risk evaluated and minimized;
then a statistics is constructed approximating the best limit solution.

A series of asymptotically unimprovable
solutions is obtained for a number of most popular many-dimensional
statistical problems: estimating expectation vectors,
regression, discriminant analysis, and the solution of
empiric systems of linear algebraic equations.

These improved essentially
multiparametric solutions have a number of practical
advantages:

--they are non-degenerating and always stable;

--they are approximately unimprovable in wide classes;

--their optimality is approximately independent of distributions;

--in case of bounded number of parameters,
these solutions pass to the conventional consistent ones.