Mathematical modeling on the base of functions density of normal distribution

Iuliia   Pinkovetskaia; Yulia  Nuretdinova; Ildar  Nuretdinov; Natalia   Lipatova

doi:10.46925//rdluz.33.04

Iuliia Pinkovetskaia Department of Economic Analysis and State Management, Ulyanovsk State University http://orcid.org/0000-0002-8224-9031
Yulia Nuretdinova Department of Economic Security, Accounting and Audit, Ulyanovsk State University http://orcid.org/0000-0002-2356-4162
Ildar Nuretdinov Department of Finance and Credit, Ulyanovsk State Agrarian University named after P. A. Stolypin, Ulyanovsk, 432600, Russia http://orcid.org/0000-0003-2511-4031
Natalia Lipatova Department of Economic Theory and Economics of Agriculture, Samara State Agrarian University, Kinel, 446430, Russia http://orcid.org/0000-0002-3167-7271

DOI: https://doi.org/10.46925//rdluz.33.04

Keywords: Mathematical modeling; normal distribution functions; statistical tests; regions; indicators

Abstract

One of the urgent tasks in many modern scientific studies is the comparative analysis of indicators that characterize large sets of similar objects located in different regions. Given the significant differences between the regions compared, this analysis should be carried out using relative indicators. The objective of the study was to use the density functions of the normal distribution to model empirical data that describe the compared sets of objects located in different regions. The methodological approach was based on the Chebyshev and Lyapunov theorems. The research results focus on the main stages of the construction of normal distribution functions and the corresponding histograms, as well as the determination of the parameters of these functions. The work possesses a degree of originality, since it provides answers to questions such as the justification of the necessary information base; performing computational experiments and developing alternative options for the generation of normal distribution density functions; comprehensive evaluation of the quality of the functions obtained through three statistical tests: Pearson, Kolmogorov-Smirnov, Shapiro-Wilk; identification of patterns that characterize the distribution of indicators of the sets of objects considered. Examples of empirical data models are given using distribution functions to estimate the share of innovative firms in the total number of firms in the regions of Russia.

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Author Biographies

Iuliia Pinkovetskaia , Department of Economic Analysis and State Management, Ulyanovsk State University

Professor. Department of Economic Analysis and State Management, Ulyanovsk State University, Ulyanovsk, 432000, Russia.

Yulia Nuretdinova , Department of Economic Security, Accounting and Audit, Ulyanovsk State University

Professor. Department of Economic Security, Accounting and Audit, Ulyanovsk State University, Russia

Ildar Nuretdinov , Department of Finance and Credit, Ulyanovsk State Agrarian University named after P. A. Stolypin, Ulyanovsk, 432600, Russia

Professor. Department of Finance and Credit, Ulyanovsk State Agrarian University named after P. A. Stolypin, Ulyanovsk, 432600, Russia

Natalia Lipatova , Department of Economic Theory and Economics of Agriculture, Samara State Agrarian University, Kinel, 446430, Russia

Professor. Department of Economic Theory and Economics of Agriculture, Samara State Agrarian University, Kinel, 446430, Russia

References

Afeez B., Maxwell O., Otekunrin O., Happiness O. (2018). Selection and Validation of Comparative Study of Normality Test. American Journal of Mathematics and Statistics. 8(6), 190-201.

Allanson P. (1992). Farm size structure in England and Wales, 1939–89. Journal of Agricultural Economics, 43, 137-148.

Balaev A.I. (2014). Modelling Financial Returns and Portfolio Construction for the Russian Stock Market. International Journal of Computational Economics and Econometrics, 1/2(4), 32-81.

Dubrov A.M., Mkhitaryan V.S. & Troshin L.I. (2000). Multidimensional statistical methods. Moscow, Finance and Statistics.

Federal State Statistics Service. Science and innovation. Available at: https://rosstat.gov.ru/folder/14477?print=1 (accessed 15.01.2021).

Filatov S.V. (2008). Some questions of improving methods of complex assessment of the financial condition of the enterprise. Scientific and practical journal, Economics, Statistics and Informatics. Vestnik UMO, 3, 56-62.

Gmurman V.E. (2003). Theory of probability and mathematical statistics. Moscow, Higher School.

Heinhold I. & Gaede K. (1964). Ingenieur statistic. München; Wien: Springler Verlag.

Khodasevich G.B. (2021). Processing of experimental data on a computer. Basic concepts and operations of experimental data processing. Available at: http://opds.sut.ru/old/electronic_manuals/oed/f02.htm (accessed 15.01.2021).

Kramer H. (1999). Mathematical methods of statistics. Princeton, University Press

Kremer N.S. (2009). Probability theory and Mathematical statistics. Moscow, UNITY-DANA.

Harrison R.H. (1985). Choosing the Optimum Number of Classes in the Chi-Square Test for Arbitrary Power Levels The Indian Journal of Statistics. 47(3), 319-324.

Mathematical Encyclopedia (in 5 volumes). (1977). edited by I. M. Vinogradov. Moscow, Soviet encyclopedia.

On the development of small and medium businesses in the Russian Federation. Federal Law No. 209-FZ of 24.07.07. ConsultantPlus.

Orlov A.I. (2004). Econometrica. Moscow, Exam.

Pearson E.S., D’Agostino R.B. & Bowmann K.O. (1977). Test for departure from normality: Comparison of powers. Biometrika, 64, 231-246.

Pinkovetskaia I.S. (2015). Methodology of research of indicators of activity of entrepreneurial structures. Proceedings of the Karelian Scientific Center of the Russian Academy of Sciences, 3, 83-92.

Pinkovetskaia I.S. (2013). Entrepreneurship in the Russian Federation: genesis, state, prospects of development. Ulyanovsk, Ulyanovsk State University.

Pinkovetskaia I.S. (2012). Comparative analysis of entrepreneurial structures in Russia. Bulletin of the NGUEU, 1, 155-164.

Rahman M. & Wu H. (2013). Tests for normality: A comparative study. Far East Journal of Mathematical Sciences. April. 75(1), 143-164.

Razali N. & Yap B.W. (2011). Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. Journal of Statistical Modeling and Analytics, 2(1), 21-33.

Seier E. & Bonett D.G. (2002). A test of Normality with high uniform power. Computational statistics & Data Analysis. 40, 435-445.

Shapkin A.S. (2003). Economic and financial risks. Evaluation, management, investment portfolio. Moscow, Publishing and Trading Corporation "Dashkov & K".

Shapiro S.S. & Wilk M.B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3-4), 591-611.

Shapiro S.S., Wilk M.B. & Chen H.G. (1968). A comparative study of various tests for normality. Journal of the American Statistical Association, 63, 1343-1372.

Storm R. (1970). Probability theory. Mathematical statistics. Statistical quality control. Moscow, Mir.

Sturgess H. (1926). The choice of a classic intervals. Journal of the American Statistical Association, 21(153), 65-66.

Totmyanina K.M. (2011). Review of models of probability of default. Financial risk management, 01(25), 12-24.

Van der Waerden B.L. (1969). Mathematical Statistics. UK, George Allen & Unwin Ltd.

Vince R. (1992). The Mathematics of Money Management: Risk Analysis Techniques for Traders. New York, John Wiley & Sons.

Wentzel E.S. (2010). The theory of probability. Moscow, KnoRus.

Yap B.W. & Sim C.H. (2011). Comparisons of various types of normality tests, Journal of Statistical Computation and Simulation, 81(12), 2141-2155.

Yazici B. & Asma S. (2007). A comparison of various tests of normality. Journal of Statistical Computation and Simulation, 77(2), 175-183.