Asymptotic normality of improved weighted empirical distribution functions

Authors:
R. Maiboroda and O. Kubaichuk

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **69** (2003).

Journal:
Theor. Probability and Math. Statist. **69** (2004), 95-102

MSC (2000):
Primary 62G30; Secondary 62G20

Published electronically:
February 8, 2005

MathSciNet review:
2110908

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**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**, 10.1007/BF02390622**2.**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****3.**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****4.**A. V. Skorokhod,*Studies in the theory of random processes*, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0185620**

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Additional Information

**R. Maiboroda**

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. Kubaichuk**

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

DOI:
http://dx.doi.org/10.1090/S0094-9000-05-00617-4

Received by editor(s):
September 26, 2002

Published electronically:
February 8, 2005

Article copyright:
© Copyright 2005
American Mathematical Society