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.