Mathematics and Statistics Vol. 7(5), pp. 218 - 228
DOI: 10.13189/ms.2019.070508
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Ellipsoidal Approximation of Distributions and Its Applications

Igor Sinitsyn , Vladimir Sinitsyn ^*
Institute of Informatics Problems, Federal Research Center "Computer Science and Control", Russian Academy of Sciences, Russian Federation

ABSTRACT

Analytical methods of the mathematical statistics of random vectors and matrices based on the parametrization of the distributions are widely used. These methods permit to design practically simple software when it is possible to have the definite information about analytical properties of the distributions under research. The main difficulty in practical applications of the methods based on the parametrization of the distributions is the rapid increase of the number of equations for the moments, the semiinvariants or the coefficients of the truncated orthogonal expansions of the dimension or the state vector (extended in the general case) and the maximal order of the moments involved. The number of equations for the parameters becomes exceedingly large in such cases. For structural parametrization and/or approximation of the probability densities of the random vectors we shall apply the ellipsoidal densities, i.e. the densities whose planes of the levels of equal probability are similar concentric ellipsoids (the ellipses for two-dimensional vectors, the ellipsoids for three-dimensional vectors, the hyperellipsoids for the vectors of the dimension more than three). In particular, a normal distribution in any finite-dimensional space has an ellipsoidal structure. The distinctive characteristics of such distributions consists in the fact that their probability densities are the functions of positively determined quadratic form where is an expectation of the random vector is some positively determined matrix. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters till () where being the number of probabilistic moments. While using ellipsoidal linearization method (ELM) we get Basic EAM and ELM foundations and applications to problems of mathematical statistics and ellipsoidal distributions with invariant measure in populational Volterra differential stochastic nonlinear systems are considered.

KEYWORDS
Distributions with Invariant Measure, Ellipsoidal Approximation Method (EAM), Ellipsoidal Linearization Method (ELM), Generalized Student Distribution, Gaussian (Normal) Distribution, Wishart Distribution, Volterra Stochastic Systems

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Igor Sinitsyn , Vladimir Sinitsyn , "Ellipsoidal Approximation of Distributions and Its Applications," Mathematics and Statistics, Vol. 7, No. 5, pp. 218 - 228, 2019. DOI: 10.13189/ms.2019.070508.

(b). APA Format:
Igor Sinitsyn , Vladimir Sinitsyn (2019). Ellipsoidal Approximation of Distributions and Its Applications. Mathematics and Statistics, 7(5), 218 - 228. DOI: 10.13189/ms.2019.070508.