|
Some applications of the Gnedenko-Korolyuk method to empirical distributions
Author(s):
E.
O.
Lutsenko;
O.
V.
Marinich;
I.
K.
Matsak
Translated by:
O. I. Klesov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika,
vipusk 78
(2008).
Journal:
Theor. Probability and Math. Statist.
No. 78
(2009),
133-146.
MSC (2000):
Primary 60B12
Posted:
August 4, 2009
Retrieve article in:
PDF
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:
-
- 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)
Similar Articles:
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC
(2000):
60B12
Retrieve articles in all Journals with MSC
(2000):
60B12
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:
10.1090/S0094-9000-09-00767-4
PII:
S 0094-9000(09)00767-4
Keywords:
Empirical distribution function,
Kolmogorov theorem,
Gnedenko--Korolyuk method
Received by editor(s):
2/JUL/2007
Posted:
August 4, 2009
Copyright of article:
Copyright
2009,
American Mathematical Society
|
Comments: webmaster@ams.org