arXiv subjects

Nataliia Voinalovych

Publications and source records attributed to Nataliia Voinalovych.

3 recordsLinked to original sources

arXiv2025

Mixed exponential statistical structures and their approximation operators

The paper examines the construction and analysis of a new class of mixed exponential statistical structures that combine the properties of stochastic models and linear positive operators. The relevance of the topic is driven by the growing need to develop a unified theoretical framework capable of describing both continuous and discrete random structures that possess approximation properties. The aim of the study is to introduce and analyze a generalized family of mixed exponential statistical structures and their corresponding linear positive operators, which include known operators as particular cases. We define auxiliary statistical structures B and H through differential relations between their elements, and construct the main Phillips-type structure. Recurrent relations for the central moments are obtained, their properties are established, and the convergence and approximation accuracy of the constructed operators are investigated. The proposed approach allows mixed exponential structures to be viewed as a generalization of known statistical systems, providing a unified analytical and stochastic description. The results demonstrate that mixed exponential statistical structures can be used to develop new classes of positive operators with controllable preservation and approximation properties. The proposed methodology forms a basis for further research in constructing multidimensional statistical structures, analyzing operators in weighted spaces, and studying their asymptotic characteristics.

math.ST↗

arXiv2025

Study of power series distributions with specified covariances

This paper presents a study of power series distributions (PSD) with prescribed covariance characteristics. Such distributions constitute a fundamental class in probability theory and mathematical statistics, as they generalize a wide range of well-known discrete distributions and enable the description of various stochastic phenomena with a predetermined variance structure. The aim of the research is to develop analytical methods for constructing power series distributions with given covariances and to establish the conditions under which a particular function can serve as the covariance of a certain PSD. The paper derives a first-order differential equation for the generating function of the distribution, which determines the relationship between its parameters and the form of the covariance function. It is shown that the choice of an analytical or polynomial covariance completely specifies the structure of the corresponding generating function. The analysis made it possible to construct new families of PSDs that generalize the classical Bernoulli, Poisson, geometric, and other distributions while preserving a given covariance structure. The proposed approach is based on the analytical relationship between the generating function and the covariance function, providing a framework for constructing stochastic models with predefined dispersion properties. The results obtained expand the theoretical framework for describing discrete distributions and open up opportunities for practical applications in statistical estimation, modeling of complex systems, financial processes, machine learning where it is crucial to control the dependence between the mean and the variation. Further research may focus on constructing continuous analogues of such distributions, studying their limiting properties, and applying them to problems of regression and Bayesian analysis.

math.ST↗

arXiv2025

On a power series distribution with mean parameterization

The article examines the distribution of the power series of the function $ w(y) = \left( 1 + \sqrt{1 - y} \right)^{-\frac{1}{2}}. $ The distribution of the considered function into a power series is obtained $ \left(1 + \sqrt{1 - y}\right)^{-\frac{1}{2}} = \sum_{m=0}^{\infty} \frac{(4m)! \, 16^{-m}}{(2m)! \, (2m+1)! \, \sqrt{2}} \, y^m. $ The dispersion function is found $ ν(x) = x (2x + 1)(4x + 1), \; x > 0. $ A distribution with mean parameterization is constructed $ \Pr(ξ= k) = \binom{4k + 1}{2k} \, 2^{-k} \, x^k \, (2k + 1)^{k + \frac{1}{2}} \, (4k + 1)^{-2k - \frac{3}{2}}, \; x > 0. $ It is proved that the raw moments $α_m$, central moments $μ_m$, cumulants $χ_m, \; m = 1, 2, \ldots$ satisfy the following recurrence relations: $ α_{m+1} = x α_m + ν(x) \frac{dα_m}{dx}, \; α_0 = 1, \; α_1 = x; \quad μ_{m+1} = m μ_{m-1} + ν(x) \frac{dμ_m}{dx}, \; μ_0 = 1, \; μ_1 = 0; \quad χ_{m+1} = ν(x) \frac{dχ_m}{dx}, \; χ_1 = x. $

math.GM↗