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Some applications of the Gnedenko-Korolyuk method to empirical distributions


Authors: E. O. Lutsenko, O. V. Marinich and I. K. Matsak
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 133-146
MSC (2000): Primary 60B12
DOI: https://doi.org/10.1090/S0094-9000-09-00767-4
Published electronically: August 4, 2009
MathSciNet review: 2446854
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Abstract | References | Similar Articles | Additional Information

Abstract: A new proof of the Kolmogorov theorem on the asymptotic behavior of the deviation between a theoretical and an empirical distribution function is presented. We use the Gnedenko-Korolyuk approach based on some combinatorial properties of the merged sample constructed from two other independent samples. Some statistical applications of the Gnedenko-Korolyuk theorem are discussed.


References [Enhancements On Off] (What's this?)

  • 1. A. Kolmogorov, Sulla determinazione empirica di una legge di distribuzione, Giorn. Ist. Ital. Attuari 4 (1933), no. 1, 83-91.
  • 2. N. V. Smirnov, Theory of Probability and Mathematical Statistics, Nauka, Moscow, 1970. (Russian) MR 0265117 (42:30)
  • 3. B. V. Gnedenko and V. S. Korolyuk, On the maximal deviation between two empirical distributions, Doklady AN SSSR 80 (1951), no. 4, 525-528. (Russian) MR 0045357 (13:570l)
  • 4. V. S. Korolyuk, On the deviation between the empirical distributions for the case of two independent samples, Izv. AN SSSR, Ser. Matem. 19 (1955), no. 1, 81-96. (Russian) MR 0067418 (16:727c)
  • 5. J. L. Doob, Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statist. 20 (1949), 393-403. MR 0030732 (11:43a)
  • 6. P. Gaenssler and W. Stute, Empirical processes: A survey of results for independent and identically distributed random variables, Ann. Probab. 7 (1979), no. 2, 193-243. MR 525051 (80d:60002)
  • 7. M. Csörgö and P. Révész, Strong Approximations in Probability and Statistics, Akademiai Kiado, Budapest, 1981. MR 666546 (84d:60050)
  • 8. E. V. Khmaladze, Some applications of the theory of martingales in statistics, Uspekhi Mat. Nauk 37 (1982), no. 6, 194-212; English transl. in Russ. Math. Surv. 37 (1982), no. 6, 215-237. MR 683280 (84c:62066)
  • 9. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. MR 0233396 (38:1718)
  • 10. E. Pitman, Some Basic Theory for Statistical Inference, Chapman and Hall, London, 1979. MR 549771 (81f:62001)
  • 11. I. I. Gikhman and A. V. Skorokhod, Theory of Stochastic Processes, vol. 1, Nauka, Moscow, 1971; English. transl., Springer-Verlag, Berlin-Heidelberg-New York, 1974. MR 0341539 (49:6287)
  • 12. B. V. Gnedenko, Theory of Probability, Sixth edition, Nauka, Moscow, 1988; English. transl., Gordon and Breach Science Publishers Newark, NJ, 1997.
  • 13. P. Schmid, On the Kolmogorov and Smirnov limit theorems for discontinuous distribution functions, Ann. Math. Statist. 29 (1958), 1011-1027. MR 0101582 (21:392)
  • 14. A. A. Borovkov, Mathematical Statistics, Nauka, Moscow, 1984; English. transl., Taylor and Francis, Amsterdam, 1999. MR 782295 (86i:62001)

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Additional Information

E. O. Lutsenko
Affiliation: Department of Operation Research, Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: ievgen_lutsenko@ukr.net

O. V. Marinich
Affiliation: Department of Operation Research, Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: marinich@voliacable.com

I. K. Matsak
Affiliation: Department of Operation Research, Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: mik@unicyb.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-09-00767-4
Keywords: Empirical distribution function, Kolmogorov theorem, Gnedenko--Korolyuk method
Received by editor(s): July 2, 2007
Published electronically: August 4, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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