Estimation of the density of a distribution from data with an admixture
Authors:
N. Lodatko and R. Maĭboroda
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 73 (2006), 99-108
MSC (2000):
Primary 62G07; Secondary 62G20
DOI:
https://doi.org/10.1090/S0094-9000-07-00684-9
Published electronically:
January 17, 2007
MathSciNet review:
2213844
Full-text PDF Free Access
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Abstract: We consider the problem of estimation of a density from observations of a two-component mixture with varying concentrations. It is assumed that the distribution of the first component is unknown, while a parametric model is (perhaps) available for the second component. Applying the sieve maximum likelihood method we construct histogram-type estimators for the densities of distributions of the components and estimators for unknown parameters of the second component. We prove the consistency of the estimators and obtain estimates for the rate of convergence.
References
- A. A. Borovkov, Matematicheskaya statistika, “Nauka”, Moscow, 1984 (Russian). Otsenka parametrov. Proverka gipotez. [Estimation of parameters. Testing of hypotheses]. MR 782295
- Yu. O. Ivan’ko, Asymptotic behavior of kernel estimators of densities and their derivatives constructed from a mixture with varying concentrations, Visnyk Kyiv Nat. Univ., Ser. Mathematics and Mechanics 9–10 (2003), 29–35.
- R. E. Maĭboroda, Statistical Analysis of Mixtures. A Course of Lectures, Kyiv University, Kyiv, 2003. (Ukrainian)
- O. V. Sugakova, Asymptotics of a kernel estimate for the density of a distribution constructed from observations of a mixture with varying concentration, Teor. Ĭmovīr. Mat. Stat. 59 (1998), 156–166 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 59 (1999), 161–171 (2000). MR 1793776
- Luc Devroye and László Györfi, Nonparametric density estimation, Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics, John Wiley & Sons, Inc., New York, 1985. The $L_1$ view. MR 780746
- Xiaotong Shen, On methods of sieves and penalization, Ann. Statist. 25 (1997), no. 6, 2555–2591. MR 1604416, DOI https://doi.org/10.1214/aos/1030741085
References
- A. A. Borovkov, Mathematical Statistics, “Nauka”, Moscow, 1984; English transl., Gordon and Breach Science Publishers, Amsterdam, 1998. MR 782295 (86i:62001)
- Yu. O. Ivan’ko, Asymptotic behavior of kernel estimators of densities and their derivatives constructed from a mixture with varying concentrations, Visnyk Kyiv Nat. Univ., Ser. Mathematics and Mechanics 9–10 (2003), 29–35.
- R. E. Maĭboroda, Statistical Analysis of Mixtures. A Course of Lectures, Kyiv University, Kyiv, 2003. (Ukrainian)
- O. V. Sugakova, Asymptotics of a kernel estimate for the density of a distribution constructed from observations of a mixture with varying concentration, Teor. Ĭmovīr. Mat. Stat. 59 (1998), 156–166; English transl. in Theor. Probability and Math. Statist. 59 (1999), 161–171. MR 1793776
- L. Devroye and L. Gyorfi, Nonparametric Density Estimation. The $L_ 1$ View, Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics, John Wiley & Sons, Inc., New York, 1985. MR 780746 (86i:62065)
- X. Shen, On methods of sieves and penalization, Ann. Statist. 25 (1997), no. 6, 2555–2591. MR 1604416 (2000a:62074)
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Additional Information
N. Lodatko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
lodatko@yandex.ru
R. Maĭboroda
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
mre@univ.kiev.ua
Keywords:
Sieve maximum likelihood method,
histogram,
a mixture with varying concentrations,
consistency
Received by editor(s):
December 4, 2004
Published electronically:
January 17, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
