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100 0,89624 0,89898 0,98204 0,98260
150 0,89362 0,89477 0,98079 0,98140
200 0,89146 0,89224 0,98024 0,98070
250 0,88968 0,89058 0,97986 0,98020
300 0,88855 0,88945 0,97952 0,97987
400 0,88682 0,88752 0,97907 0,97939
500 0,88569 0,88651 0,97879 0,97908
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500 0,90421 0,90245 0,95312 0,95199
1000 0,90271 0,90227 0,95270 0,95239
2000 0,90182 0,90151 0,95232 0,95212
5000 0,90059 0,90151 0,95178 0,95166
10000 0,89989 0,89976 0,95145 0,95138
20000 0,89938 0,89931 0,95120 0,95115
30000 0,89915 0,89909 0,95107 0,95104
40000 0,89901 0,89897 0,95100 0,95097
50000 0,89892 0,89888 0,95095 0,95093
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[10,11,12,13], .

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1. .. // . 1987. . 32, 4. . 790-792.
2. .., .. // . 2011. 11. . 6-11.
3. .., .. - // . 1985. . 30, 3. . 572-573.
4. .. // . .. . . . 2003. 2. . 28-41.
5. .., .. - - // . 2010. .15, 7. . 18-26.
6. Crowder M.J. Multivariate Survival Analysis and Competing Risks. CRC Press; Chapman and Hall, 2012. 417 p. (Ser. Texts in Statistical Science).
7. May S., Hosmer D.W. A simplified method of calculating an overall goodness-of-fit test for the Cox proportional hazards model // Lifetime Data Analysis. 1998. Vol. 4, no. 2. P. 109120. DOI: 10.1023/A:1009612305785
8. .. // - , .. . . . . 3. .: . .. , 2009. . 227-230.
9. Corder G.W., Foreman D.I. Nonparametric statistics: A step-by-step approach. New Jersey: Wiley, 2014. 288 p.
10. .., .. . .: , 1983. 416 .
11. Hajek J., Sidak Z. Theory of rank tests. London: Academic Press, 2004. 438 p.
12. Gao J., Ozturk O. Two-sample distribution-free inference based on partially rank-ordered set samples // Statistics and Probability Letters. 2012. Vol. 82, iss. 5. P. 876-884. DOI: 10.1016/j.spl.2012.01.021
13. McLain A.C., Ghosh S.K. Nonparametric estimation of the conditional mean residual life function with censored data // Lifetime Data Analysis. 2011. Vol. 17, no. 4. P. 514-532. DOI: 10.1007/s10985-011-9197-x
Science and Education of the Bauman MSTU, 2014, no. 11, pp. 227-237.
DOI: 10.7463/1114.0740251
Received: Revised:
05.11.2014 25.11.2014
Science ^Education
of the Bauman MSTU
ISSN 1994-0448 Bauman Moscow State Technical Unversity
The Method of Calculating the Exact Distributions of the Kolmogorov-Smirnov Statistics in Case of Violation of Homogeneity and Independence of the Analyzed Samples
N.D. Tiannikova1' , V.I.Timonin1 "tianHikova@yandexiu
:Bauman Moscow State Technical University, Moscow, Russia
Keywords: the Kolmogorov-Smirnov statistics, non-parametric statistics, Kaplan-Meier estimates,
power Lehmann dependences
To establish the relationship between the distribution functions of the experimental results for different values of the external factors are most commonly used parametric models in which the parameters of the distribution functions depend on factors, and their views do not change. Meanwhile, when we have a small amount of data (and this is more common in practice), the distribution function is often unknown, and it is difficult to determine. Hence, it is of great importance to evaluate different relationships between the distribution laws without specifying a particular form of the distribution (these issues are handled by non-parametric statistics). The most common model, used to establish the relationship between the theoretical distribution functions of different samples, is the Cox model. Furthermore, even for testing the homogeneity of multiple samples, experiments, which are necessary to obtain them, are so complex that the obtained samples are dependent. So, all of these tasks requires the development of new non-parametric tests for dependency. Due to the fact, that the volume of the sample is always small, knowledge of exact distributions of statistics, which are used, is of special importance. The paper develops a general method for tabulating the exact distributions (for finite volumes of samples) of a wide class of statistics of the Kolmogorov-Smirnov test. With the appropriate specialization of the proposed algorithm, it allows us to calculate the distribution of various statistics of the specified type. In particular, it is applicable for calculating the distribution of statistics such as Kiefer-Gikhman used to check the dependencies between Lehmann distribution functions of several samples. With small modifications it allows us to tabulate the distribution statistics of the Kolmogorov-Smirnov used for checking the homogeneity of dependent samples. Along with the fact that the method has great generality, it also allows us to calculate the exact distribution for very large volumes of samples. This fact allows us to estimate the volume of the sample, in which the asymptotic distribution can be applied.
The limits of this method applicability are also given. It assumes the validity of a special
model of random movement of particle on a multidimensional lattice in which the future behavior of the particle trajectory at presently given is independent of its past.
References
1. Timonin V.I. On the Limit Distribution of Statistics of a Nonparametric Test. Teoriia veroiatnostei i eeprimeneniia, 1987, vol. 32, no. 4, pp. 790-792. (English translation: Theory of Probability and Its Applications, 1988, vol. 32, no. 4, pp. 721-724. DOI: 10.1137/1132108 ).
2. Ermolaeva M.A., Timonin V.I. A multi-Sample Analogue to the Smirnov Test Criterion for the Lehmann Power Hypothesis. Elektromagnitnye volny i elektronnye sistemy = Electromagnetic Waves and Electronic Systems, 2011, no. 11, pp. 6-11. (in Russian).
3. Timonin V.I., Chernomordik O.M. A Method for Calculating the Exact Distribution of Kol-mogorov-Smirnov Statistics under Lehmann Alternatives. Teoriya veroyatnostey i ee primenenie, 1985, vol. 30, no. 3, pp. 572-573. (English translation: Theory of Probability and Its Applications, 1986, vol. 30, no. 3, pp. 608-610. DOI: 10.1137/1130077 ).
4. Timonin V.I. Optimization of Preliminary Studies in Theory of Forced Testing. Vestnik MGTU im. N.E. Baumana. Ser. Estestvennye nauki = Herald of the Bauman MSTU. Ser. Natural science, 2003, no. 2, pp. 28-41. (in Russian).
5. Timonin V.I., Ermolaeva M.A. About Kaplan-Meyer Estimators in Statistics Similar to Kol-mogorov-Smirnov for Testing the Hypothesis in Variable Load Tests. Elektromagnitnye volny i elektronnye sistemy = Electromagnetic Waves and Electronic Systems, 2010, vol. 15, no. 7, pp. 18-26. (in Russian).
6. Crowder M.J. Multivariate Survival Analysis and Competing Risks. CRC Press; Chapman and Hall, 2012. 417 p. (Ser. Texts in Statistical Science).
7. May S., Hosmer D.W. A simplified method of calculating an overall goodness-of-fit test for the Cox proportional hazards model. Lifetime Data Analysis, 1998, vol. 4, no. 2, pp. 109-120. DOI: 10.1023/A:1009612305785
8. Ermolaeva M.A. Non-parametric analysis of the relationship between the distributions of operating time to failure of products and devices in different operating conditions. Trudy rossiyskogo nauchno-tekhnicheskogo obshchestva radiotekhniki, elektroniki i svyazi imeni A.S. Popova. Ser. Akustoopticheskie i radiolokatsionnye metody izmereniy i obrabotki informatsii. Vyp. 3 [Proc. of the Russian Scientific and Technical Society of Radio Engineering, Electronics and Communication named after A.S. Popov. Ser. Acoustooptical and Radar
Methods for Information Measurements and Processing. Iss. 3]. Moscow, RNTORES Publ., 2009, pp. 227-230. (in Russian).
9. Corder G.W., Foreman D.I. Nonparametric statistics: A step-by-step approach. New Jersey, Wiley, 2014. 288 p.
10. Bol'shev L.N., Smirnov N.V. Tablitsy matematicheskoy statistiki [Tables of Mathematical Statistics]. Moscow, Nauka Publ., 1983. 416 p. (in Russian).
11. Hajek J., Sidak Z. Theory of rank tests. London, Academic Press, 2004. 438 p.
12. Gao J., Ozturk O. Two-sample distribution-free inference based on partially rank-ordered set samples. Statistics and Probability Letters, 2012, vol. 82, iss. 5, pp. 876-884. DOI: 10.1016/j.spl.2012.01.021
13. McLain A.C., Ghosh S.K. Nonparametric estimation of the conditional mean residual life function with censored data. Lifetime Data Analysis, 2011, vol. 17, no. 4, pp. 514-532. DOI: 10.1007/s10985-011-9197-x