Some estimators of the probability density function in spaces of arbitrary nature are used for various tasks in statistics of non-numerical data. Systematic exposition of the theory of such estimators had a start in our work [2]. This article is a direct continuation of the article [2]. We will regularly use references to conditions and theorems of the article [2], in which we introduced several types of nonparametric estimators of the probability density. We studied more linear estimators. In this article we consider particular cases - kernel density estimates in spaces of arbitrary nature. When estimating the density of the one-dimensional random variable, kernel estimators become the Parzen-Rosenblatt estimators. Asymptotic behavior of kernel density estimators in the general case of an arbitrary nature spaces are devoted to Theorem 1 - 8. Under different conditions we prove the consistency and asymptotic normality of kernel density estimators. We have studied uniform convergence. We have introduced the concept of "preferred rate differences" and studied nuclear density estimators based on it. We have also introduced and studied natural affinity measures which are used in the analysis of the asymptotic behavior of kernel density estimators. We have found the asymptotic behavior of dispersions of kernel density estimators and considered the examples including kernel density estimators in finite-dimensional spaces and in the space of square-integrable functions
The article is devoted to the nonparametric point and
interval estimation of the characteristics of the
probabilistic distribution (the expectation, median,
variance, standard deviation, variation coefficient) of
the sample results. Sample values are regarded as the
implementation of independent and identically
distributed random variables with an arbitrary
distribution function having the desired number of
moments. Nonparametric analysis procedures are
compared with the parametric procedures, based on
the assumption that the sample values have a normal
distribution. Point estimators are constructed in the
obvious way - using sample analogs of the
theoretical characteristics. Interval estimators are
based on asymptotic normality of sample moments
and functions from them. Nonparametric asymptotic
confidence intervals are obtained through the use of
special output technology of the asymptotic relations
of Applied Statistics. In the first step this technology
uses the multidimensional central limit theorem,
applied to the sums of vectors whose coordinates are
the degrees of initial random variables. The second
step is the conversion limit multivariate normal
vector to obtain the interest of researcher vector. At
the same considerations we have used linearization
and discarded infinitesimal quantities. The third step
- a rigorous justification of the results on the
asymptotic standard for mathematical and statistical
reasoning level. It is usually necessary to use the
necessary and sufficient conditions for the
inheritance of convergence. This article contains 10
numerical examples. Initial data - information about
an operating time of 50 cutting tools to the limit
state. Using the methods developed on the
assumption of normal distribution, it can lead to
noticeably distorted conclusions in a situation where
the normality hypothesis failed. Practical
recommendations are: for the analysis of real data we
should use nonparametric confidence limits
In this paper, we consider Einstein's theory of gravitation in connection with Yang-Mills theory. The model of the metric satisfying the basic requirements of quantum theory is proposed. The mechanism of generation of baryonic matter of dark energy is discussed
The article presents a new approach to 2D modeling of transport of salt ions in EMC (electro systems: electrodialysis devices, electro-cells, etc.) under the condition of electrical neutrality with limiting and overlimiting current density. For definiteness as seen half of EMS channel EDA desalting (electrodialysis apparatus), the right border, which serves as a CEM (cation exchange membrane). The new approach in the use of partial differential equations of the first order, instead of equations of convective diffusion. A common method of transport modeling binary electrolyte in the EMS under the condition of electrical neutrality, is to use the equation of convective diffusion (partial differential equations of the second order). The article presents a new approach to modeling 2D transfer binary electrolyte in EMS under the same conditions, using partial differential equation of the first order for the decision, which does not require a boundary condition for concentration on the membrane surface. This allows you to simulate the transport of salt ions, as in prelimit and exorbitant current density and to determine the boundaries of the field of electrical neutrality
We have considered the basic mathematical tools (theorems, methods) which are used regularly in the justification of new results in the field of statistical methods: rules of large numbers, central limit theorems, the necessary and sufficient conditions for the inheritance of convergence, the linearization method, the invariance principle
The article presents a project of the capacitor in the
Yang-Mills theory. Model capacitor represents the
equipotential surfaces separated by a space. To
describe the mechanism of condensation
chromodynamics field used numerical models
developed based on an average of the Yang-Mills
theory. In the present study, we used eight-scalar
component model that in the linear case is divided
into two groups containing three or five fields
respectively. In contrast to classical electrodynamics,
a static model of the Yang-Mills is not divided into
independent equations because of the nonlinearity of
the model itself. However, in the case of a linear
theory separation is possible. It is shown that in this
particular case, the Yang-Mills theory is reduced to
Poisson theory, which describes the electrostatic and
magnetostatic phenomena. In the present work it is
shown that in a certain region of the parameters of the
capacitor of the Yang-Mills theory on the functional
properties of the charge accumulation and retention of
the field is similar to the capacitor of the electrostatic
field or a magnet in magnetostatics. This means that
in nature there are two types of charges, which are
sources of macroscopic Yang-Mills field, which are
similar to the properties of electric and magnetic
charges in the Poisson theory. It is shown that in
Yang-Mills only one type of charge may be
associated with the distribution density of the
substance, while another type of charge depends on
the charge distribution of the first type. This allows us
to provide an explanation for the lack of symmetry
between electric and magnetic charges
The metric of inhomogeneous rotating Universe is discussed. There are examples of universal metrics obtained in Einstein's theory of gravitation. On the basis of solutions of Einstein’s equation we have proposed universal metric describing the properties of galaxies, groups and clusters of galaxies in inhomogeneous rotating Universe
In this article we consider Einstein's theory of gravity in relation to the Yang-Mills theory. It is shown that in Einstein's theory there exists a metric together with the Yang-Mills theory, in which the field equations are reduced to the Liouville equation describing the
evolution of stars. The mechanism of generation of stellar energy of dark energy in the processes of geometric turbulence is discussed
The probabilistic model of grouping data (including multidimensional data) is described. We have also generalized Euler-Maclaurin’s formulas. With its help Sheppard’s corrections and corrections on
grouping for correlation coefficient are received. We have found and studied asymptotical corrections on grouping data generally. Accuracy of approach has been estimated
In this work it is investigated the propagation of unsteady-state waves in a vicoelastic semi space for the arbitary heterogeneous functions at the different boarded conditions with the help of Laplas
transformation and exponentional Truriers conversion. The received solution has been analyzed in a private case when medium properties are describing by Rzhanitsin’s core