STATISTICAL SIMULATIONS METHOD IN APPLIED STATISTICS

The new paradigm of mathematical research methods is based on the effective application of information and communication technologies both in calculating the characteristics of the methods of data analysis and in simulation modeling. Pseudo-random number generators underlie many modern data analysis technologies. To solve specific applied problems, researchers permanently develop the new methods for processing statistical data, i.e., measurement results (observations, tests, analyzes, experiments) and expert estimations. The properties of each newly proposed method must be studied. The intellectual tools are limit theorems and method of statistical simulations (Monte-Carlo method). In 2016, our journal opened a discussion on the current state and prospects for the development of statistical modeling, i.e., the theory and practice of applicating the method of the statistical simulations (Monte-Carlo method), and various variants of the simulation. The previous discussion about the properties of such generators was conducted in our journal in 1985 - 1993. This article is devoted to application of the statistical simulations method to the study of the properties of statistical criteria for testing the homogeneity of two independent samples. We consider: the Kramer - Welch criterion, which coincides with Student's criterion when sample sizes are equal; the criteria of Lord, Wilcoxon (Mann - Whitney), Wolfowitz, Van der Waerden, Smirnov, со 2 (Lehmann - Rosenblatt). It is necessary to set the distribution functions of the elements of two samples. We use the normal and Weibull - Gnedenko distributions. It is shown advisable to use the Lehmann - Rosenblatt со 2 test when testing the hypothesis of coincidence of the distribution functions of two samples. If there is a reason to assume that the distributions differ mainly in the shift, then the Wilcoxon and Van der Waerden criteria can be used. However, even in this case, the со 2 test may be more powerful. In the general case, apart from the Lehmann - Rosenblatt criterion, the use of the Smirnov criterion is permissible, taking into account the difference between the real level of significance and the nominal one. The frequency of the discrepancies of statistical findings based on different criteria is studied.

applied statistics, statistical simulations method, Monte-Carlo method, pseudo-random number generators, criteria for testing statistical hypotheses, homogeneity of two independent samples, computational experiment, Kramer - Welch criterion, Lord criterion, Wilcoxon criterion, Van der Waerden criterion, Smirnov criterion, Lehmann - Rosenblatt criterion

1. Gorskii V G., Orlov A. I. Mathematical methods of research: results and prospects / Zavod. Lab. Diagn. Mater. 2002. Vol. 68. N 1. E 108 - 112 [in Russian].

2. Orlov A. I. The new paradigm of Applied Statistics / Zavod. Lab. Diagn. Mater. 2012. Vol. 78. N 1. E 87 - 93 [in Russian].

3. Orlov A. I. On the new paradigm of mathematical research methods / Nauch. Zh. KubGAU. 2016. N 122. E 807 - 832 [in Russian].

4. Orlov A. I., Lucenko E. V. System fuzzy interval mathematics. — Krasnodar: KubGAU, 2014. — 600 p. [in Russian].

5. Orlov A. I. Development of the Methods of mathematical Research (2006-2015) / Zavod. Lab. Diagn. Mater. 2017. Vol. 83. N 1. Fart 1. E 78 - 86 [in Russian].

6. Kolmogorov A. N. Information Theory and Algorithm Theory. — Moscow: Nauka, 1987. — 304 p. [in Russian].

7. Grigor'ev Yu. D. Monte Carlo method: accuracy of asymptotic solutions and Quality of Fseudorandom number generators / Zavod. Lab. Diagn. Mater. 2016. Vol.82. N7. E 72 - 84 [in Russian].

8. Orlov A. I. Limit theorems and the Monte Carlo method / Zavod. Lab. Diagn. Mater. 2016. Vol.82. N7. E 67 - 72 [in Russian].

9. Kutuzov O. I., Tatarnikova Т. M. Fractical Experience of Using Monte Carlo method / Zavod. Lab. Diagn. Mater. 2017. Vol. 83. N 3. E 65 - 70 [in Russian].

10. Aronov I. Z., Maksimova O. V. Analysis of time to reach consensus on the work of the technical committees of standardization as a results of statistical modeling / Zavod. Lab. Diagn. Mater. 2017. Vol. 83. N 3. E 71 - 77 [in Russian].

11. Orlov A. I. Consensus and truth (commens to the article published by I. Z. Aronov and O. V Maximova) / Zavod. Lab. Diagn. Mater. 2017. Vol. 83. N 3. E 78 - 79 [in Russian].

12. Orlov A. I. Importance of information and communication technologies for mathematical methods of research / Zavod. Lab. Diagn. Mater. 2017. Vol. 83. N 7. E 5 - 6 [in Russian].

13. Zhukov M. S. On the algorithms for Kemeny median calculation / Zavod. Lab. Diagn. Mater. 2017. Vol. 83. N 7. E 72 - 78 [in Russian].

14. Gadolina I. V, Lisachenko N. G. Development of the Method of constructing confidence intervals for percentiles of composites strength random variable using bootstrap simulation / Zavod. Lab. Diagn. Mater. 2017. Vol. 83. N 11. E 73 - 77 [in Russian].

15. Orlov A. I. Test of uniformity of two independent samplings / Zavod. Lab. Diagn. Mater. 2003. Vol. 69. N 1. E 55 - 60 [in Russian].

16. Bol'shev L. N., Smirnov N. V. Tables of mathematical statistics. 3 ed. — Moscow: Nauka, 1983. — 416 p. [in Russian].

17. Orlov A. I. Which hypothesis can be tested using the twosample Wilcoxon Criterion / Zavod. Lab. Diagn. Mater. 1999. Vol. 65. N 1. E 51 - 55.

18. Orlov A. I. Consistent tests of absolute homogeneity of independent sampling / Zavod. Lab. Diagn. Mater. 2012. Vol. 78. N11. E 66-70.

19. Lehmann E. L. Consistency and unbiasedness of certain nonparametric tests / Ann. Math. Statist. 1951. Vol. 22. N 2. E 165 - 179.

20. Rosenblatt M. Limit theorems associated with variants of the von Mises statistic / Ann. Math. Statist. 1952. Vol. 23. N 4. E 617-623.

21. Orlov A. I. Organizational and economic modeling: a textbook: in 3 parts. Fart 3. Statistical data analysis methods. — Moscow: Izd. MGTU im. N. E. Baumana, 2012. — 624 p. [in Russian].

22. Hajek Ja., Sidak Zb. Theory of rank tests. — Frague: Academia. Fublishing House of the Czechoslovak Academy of Sciences, 1967. — 376 p.

23. Hollander M., Wolfe D. A. Nonparametric statistical methods. — New York-London-Sydney-Toronto: John Wiley and Sons, 1973. — 503 p. [in Russian].

24. Pardzhanadze A. M., Khmaladze E. V On the asymptotic theory of statistics from sequential ranks / Teor. Veroyat. Ее Frimen. 1986. Vol. XXXI. N 4. E 758 - 772 [in Russian].

25. Orlov A. I. Real and nominal significance levels in statistical hypothesis testing / Nauch. Zh. KubGAU. 2015. N 114. E 42 -54 [in Russian].

26. Forsythe G. E., Malkolm M. A., Moler С. B. Computer methods for mathematical computations. — New Jersey: Frentice-Hall, Inc., Englewood Cliffs, 1977. — 192 p.

27. Shannon R. E. Systems Simulation: the Art and Science. — New Jersey: Frentice-Hall, Inc., Englewood Cliffs, 1975. — 420 p.

28. Orlov A. I. The First World Congress of the Bernoulli Society for Mathematical Statistics and Probability Theory / Zavod. Lab. Diagn. Mater. 1987. Vol. 53. N 3. E 90 - 91.

29. Ermakov S. M. Monte Carlo method and related questions. — Moscow: Nauka, 1971. — 328 p. [in Russian].

30. Ermakov S. M., Mikhailov G. A. Statistical modeling. — Moscow: Nauka, 1982. — 296 p. [in Russian].

31. Zhurbenko I. G., Kozhevnikova I. A., Klindukhova O. V Determining the critical length of a sequence of pseudo-random numbers / Probabilistic-statistical research methods. — Moscow: MGU im. M. V Lomonosova, 1983. E 18 - 39 [in Russian].

32. Zhurbenko I. G., Kozhevnikova I. A., Smirnova O. S. On the construction and study of pseudo-random sequences by various methods / Zavod. Lab. Diagn. Mater. 1985. Vol. 51. N 5. E47-51 .

33. Zhurbenko I. G. Analysis of stationary and homogeneous random systems. — Moscow: MGU im. M. V Lomonosova, 1987. — 240 p. [in Russian].

34. Rydanova G. V A technique for studying temporal dependencies in pseudo-random number sequences / Zavod. Lab. Diagn. Mater. 1986. Vol. 52. N 1. E 56 - 58.

35. Orlov A. I. Probabilistic-statistical modeling of the interferences generated by electric locomotives / Nauch. Zh. KubGAU. 2015. N 106. E 225 - 238.

36. Orlov A. I. Theory of lucians / Nauch. Zh. KubGAU. 2014. N 101. E 275 - 304.

37. Orlov A. I. On the real possibilities of bootstrap as a statistical method / Zavod. Lab. Diagn. Mater. 1987. Vol. 53. N 10. E 82 -85.

38. Aivazyan S. A., Enyukov I. S., Meshalkin L. D. Application statistics. Basics of modeling and primary data processing. — Moscow: Finansy i statistika, 1983. — 472 p. [in Russian].

39. Hastings N. A. J., Peacock J. B. Statistical distributions. A handbook for students and practitioners. — London: Butterworth and Co (Publishers) Ltd, 1975. — 95 p.

40. Fomin V N. Rationing reliability indicators. — Moscow: Izd. Standartov, 1986. — 140 p. [in Russian].

41. Cox D. R., Hinkley D. V. Theoretical statistics. — London: Chapman and Hall, 1974. — 522 p.

42. Orlov A. I. Stable mathematical methods and models / Zavod. Lab. Diagn. Mater. 2010. Vol. 76. N 3. E 59 - 67 [in Russian].