A mixture of probability distributions is a mathematical model that allows to describe heterogeneous data. The task of separating mixtures or decomposition is the task of estimating the unknown parameters of miscible distributions. Despite the adequacy of the description of heterogeneous data, the decomposition of mixtures is a separate problem, due to the large number of parameters to be evaluated. The article carries out historical periodization,systematization, and a critical comparative analysis of existing methods and algorithms for decomposition of mixtures of probability distributions, identifies the possibilities and limitations of their application for the analysis of real populations. Based on existing algorithms, a method for separating mixtures of an arbitrary known number of probability distributions and a further typological grouping of real socio-economic aggregates is proposed. Unlike existing methods, a method for calculating threshold values to determine the boundaries of types and the number of components of the mixture, in cases where it is unknown, is proposed. Based on the proposed methodology, a typology of the subjects of the Russian Federation by the level of unemployment in the Russian Federation is carried out.

The article summarizes research results of the study on the problem of assessing the differentiation level of socio-economic development of territorial units of the Russian Federation. The authors propose an approach to measuring differentiation using mixtures of probability distributions.This technique was developed and tested on real data that allows one to determine the presence or absence of interregional differentiation. The research hypothesis, that interterritorial differentiation is estimated by a specific statistical indicator selected based on a content, qualitative analysis, served as a theoretical platform of this methodology. Differentiation is practically absent if the entire statistical population is described by a single law of probability distribution. If the statistical population is described by a mixture of probability distributions, then one should expect the presence of a significant level of differentiation by the considered indicator.In mathematical statistics, the problem of separating a mixture of probability distributions (estimating parameters of distribution densities and weighting coefficients) is traditionally solved using several similar methods. For example, the expectation-maximization (EM) algorithm, median modifications of the EM-algorithm, SEM-algorithm, taking into account the specifics of the selected object (constituent entities of the Russian Federation a small sample). To solve this problem, the authors used the SEM algorithm. As the information base of the empirical study, official statistics were used (open data from the Federal State Statistics Service).The typologies of the constituent entities of the Russian Federation were identified based on two characteristics within 2005-2017-time interval. The first one being the level of violence (using the “homicide rate” indicator the number of homicides and attempted murders per 100000 population). And second average per capita income, which made it possible, among other things, to additionally test the hypothesis of the traditional use of differentiation trends in the level of violence as an indicator of economic inequality. According to the authors, the results of this study can be used as instrumental and informational support for managerial decisions aimed at regulating the differentiation of Russian regions by the level of violence and economic inequality.

We study the model of interregional trade under monopolistic competition of producers. We obtain a local comparative statics of symmetric social optimality with respect to transport costs. Particular attention is paid to situations of free trade and autarky. For the case of two regions, counterintuitive results we obtain that (1) with low transport costs in one of the regions, public welfare can either increase or decrease; (2) when transport costs are high, trade liberalization worsens public welfare in one region and improves it in another.

The authors studied heterogeneous samples (for a number of statistical populations) taken from finite mixtures of probability distribu- tions, which reflect a number of socio-economic characteristics of the Russian society, using the information resources of the Federal State Statistics Service; in this case the number of mixing distributions (components), as well as their corresponding weights and parameters, may be unknown. In terms of content, the problem of separation (decomposition) of mixtures can be reduced to estimating unknown parameters of mixing distributions and their weights.The article considers and discusses methods for solving this problem, their advantages and disadvantages, conditions and areas of application. In the study the decomposition of mixtures of probability distributions on the population of subjects of the RussianFederation was carried out according to three interrelated statistical indicators – the unemployment rate, the poverty rate, the level of violence. These indicators can be considered as statistical measure of achieving sustainable development goals at the regional level and, at the same time, measures of socio-economic «health» of territory. Typologies of subjects of the Russian Federation were carried out according to the listed indicators. A cross-comparison of the results of the obtained typological groupings was performed, which also made it possible to identify the subjects of the Russian Federation that differ in both negative and positive trends in the context of the indicators under consideration.The authors underline that the results of the study can be used by the authorities to develop specific measures for the socio-economic development of Russia and its regions.

We study a homogeneous model of interregional trade under monopolistic competition of producers with additively separable utility and linear production costs. The mass of firms (“the length of the product line”) is determined endogenously, from conditions of free entry. We obtain the local comparative statics of the market equilibrium with respect to transport costs of “iceberg types”. Particular attention is paid to situations of free trade and autarky. We establish the following counter-intuitive result: under high transport costs, trade liberalization can decrease public welfare.