Probabilistic Mixtures in Measurements of Interterritorial Differentiation

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.

1. Gilinskiy Ya.I. Social Violence: Monograph. Saint-Petersburg: Ltd Publishing House «Alef-Press»; 2013. P. 185. (In Russ.)

2. Gilinskiy Y. Crime in Contemporary Russia. European Journal of Criminology. 2006:3(3);259-292.

3. Olkov S. The Effect of the Inequality Degree in the Distribution of Population Incomes on the Intentional Killings Level on the Planet. Study of the Laws of Distribution, Concentration and Differentiation of Inequality and Intentional Killings on Earth at the Beginning of the XXI Century. Public and Private Law. 2010:3(7);7-24. (In Russ.)

4. Olkov S., Mukhametzyanov I.Sh., Alekseev S.L., Epikhin A.Yu. (eds.) Correlation analysis of the crime structure in its explanation and forecasting, the study of the unemployment impact on the various structural components of crime in Russia. Scientific Works Bulletin of the Law Faculty «Jurist»; 2015. 181 p. (In Russ.)

5. Badov A. Geocriminogenic Situation as a Crime Factor. Proceedings of the Russian Academy of Sciences. Geographical series. 2009;(2):48-51. (In Russ.)

6. Badov A.D. The Geography of Crime in Russia: Changes for the Post-Soviet Period. Vestnik Moskovskogo universiteta. Seriya 5, Geografiya. 2009;(2):64-70. (In Russ.)

7. Luneev V.V. Crime in the XX century: Global, Regional and Russian Trends. 2nd Ed., Rev. Moscow: Wolters Kluwer Publ.; 2005. 912 p. (In Russ.)

8. Lysova A.V. Homicide in Russia, Ukraine and Belarus. In: Liem M.C.A., Pridemore W.A. (eds). Handbook of European Homicide Research: Patterns, Explanations and Country Studies. New York: Springer-Verlag; 2012. P. 251-469.

9. Land K., McCall P.L., Cohen L.E. Structural Covariates of Homicide Rates: Are There Any Invariances Across Time and Social Space? The American Journal of Sociology. 1990;95(4):922-963.

10. Tcherni M. Structural Determinants of Homicide: The Big Three. Journal of Quantitative Criminology. 2011;27(4):475-496.

11. Buonanno P., Vargas J.F. Inequality, Crime, and the Long Run Legacy of Slavery. Journal of Economic Behavior and Organization. 2019;159(C):539-552.

12. Choe J. Income Inequality and Crime in the United States. Economics Letters. 2008;101(1):31-33.

13. Dahlberg M., Gustavsson M. Inequality and Crime: Separating the Effects of Permanent and Transitory Income. Oxford Bulletin of Economics and Statistics. 2008;70(2):129-153.

14. Korolev V. The EM Algorithm, Its Modifications, and Their Application to the Problem of Separating Probability Distributions Mixtures. Theoretical Review. Мoscow: IPI RAN; 2007, 94 p. (In Russ.)

15. Glinskiy V.V., Tret’yakova O.V., Skripkina T.B. Typology of Regions of the Russian Federation by Health Care Effectiveness Level. Voprosy Statistiki. 2013;(1):57-68. (In Russ.)

16. Glinskiy V.V., Serga L.K., Pulyaevskaya V.L. Statistical Tools in Solving the Problems of Managing the Development of the Territories. Voprosy Statistiki. 2014;(10):14-20. (In Russ.)

17. Glinskiy V.V. Small Business Mythological Statistic. Problems of Turbulent Sets Study. ECO. 2008;3(9):51-62. (In Russ.)

18. Ismaiylova Yu. The Method of Moments as a Way of Probability Distributions Mixtures’ Decomposition. Statistics the Language of Digital Civilization. In: Collection of Reports of the II Open Russian Statistical Congress, 2018. P. 135-142. (In Russ.)

19. Ismaiylova Yu.N. On the Separating Mixtures of Probability Distributions by Means of the Method of Moments. Accounting and Statistics. 2018;4(52):45-51. (In Russ.)