Asymptotic normality of improved weighted empirical distribution functions
Authors:
R. Maĭboroda and O. Kubaĭchuk
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 69 (2004), 95-102
MSC (2000):
Primary 62G30; Secondary 62G20
DOI:
https://doi.org/10.1090/S0094-9000-05-00617-4
Published electronically:
February 8, 2005
MathSciNet review:
2110908
Full-text PDF Free Access
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Abstract: Weighted empirical distribution functions are often used to estimate the distributions of components in a mixture. However, weighted empirical distribution functions do not possess some properties of probability distribution functions in the case of negative weight coefficients. We consider a method allowing one to improve weighted empirical distribution functions and obtain an estimator that is a distribution function. We prove that this estimator is asymptotically normal. The limit distribution of the improved weighted empirical distribution function coincides with that of the initial estimator.
References
- R. Ē. Maĭboroda, Estimation of the distributions of the components of mixtures having varying concentrations, Ukraïn. Mat. Zh. 48 (1996), no. 4, 558–562 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 48 (1996), no. 4, 618–622 (1997). MR 1417019, DOI https://doi.org/10.1007/BF02390622
- R. Ē. Maĭboroda, Correlation analysis of mixtures. I, Teor. Ĭmovīr. Mat. Stat. 54 (1996), 99–108 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 54 (1997), 105–114. MR 1644590
- V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR 1743716
- A. V. Skorokhod, Studies in the theory of random processes, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. Translated from the Russian by Scripta Technica, Inc. MR 0185620
References
- R. E. Maĭboroda, Estimation of the distributions of the components of mixtures having varying concentrations, Ukrain. Matem. Zh. 48 (1996), no. 4, 562–566; English transl. in Ukrainian Math. J. 48 (1996), no. 4, 618–622. MR 1417019 (97j:62055)
- ---, Correlation analysis of mixtures. I, Teor. Imovir. Matem. Statist. 54 (1996), 99–108; English transl. in Theor. Probability Math. Statist. 54 (1997), 105–114. MR 1644590 (99k:62102)
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev, 1998; English transl., AMS, Providence, RI, 2000. MR 1743716 (2001g:60089)
- A. V. Skorokhod, Studies in the Theory of Random Processes, Kiev University, Kiev, 1961; English transl., Addison-Wesley, New York, 1965. MR 0185620 (32:3082b)
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Additional Information
R. Maĭboroda
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty of Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
mre@mechmat.univ.kiev.ua
O. Kubaĭchuk
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty of Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
linsta@akcecc.kiev.ua
Received by editor(s):
September 26, 2002
Published electronically:
February 8, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
