Estimation of the density of a distribution from data with an admixture

Authors:
N. Lodatko and R. Maiboroda

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **73** (2005).

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

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Abstract | References | Similar Articles | Additional Information

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.

**1.**A. A. Borovkov,*\cyr Matematicheskaya statistika*, “Nauka”, Moscow, 1984 (Russian). \cyr Otsenka parametrov. Proverka gipotez. [Estimation of parameters. Testing of hypotheses]. MR**782295****2.**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.**3.**R. E. Maiboroda,*Statistical Analysis of Mixtures. A Course of Lectures*, Kyiv University, Kyiv, 2003. (Ukrainian)**4.**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****5.**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 𝐿₁ view. MR**780746****6.**Xiaotong Shen,*On methods of sieves and penalization*, Ann. Statist.**25**(1997), no. 6, 2555–2591. MR**1604416**, https://doi.org/10.1214/aos/1030741085

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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. Maiboroda**

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

DOI:
https://doi.org/10.1090/S0094-9000-07-00684-9

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