An adaptive moment estimator of a parameter of a distribution constructed from observations with admixture

Authors:
N. Lodatko and R. Maiboroda

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **75** (2006).

Journal:
Theor. Probability and Math. Statist. **75** (2007), 71-82

MSC (2000):
Primary 62G07; Secondary 62G20

Published electronically:
January 24, 2008

MathSciNet review:
2321182

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of estimating an unknown parameter from observations with an admixture. The concentration of the admixture is varying with observations and assumed to be known, while its distribution is unknown. We study moment estimators and prove that they are consistent and asymptotically normal. We use an adaptive technique that allows us to determine estimators whose asymptotic variance is minimal among moment estimators.

**1.**A. A. Borovkov,*Matematicheskaya statistika*, “Nauka”, Moscow, 1984 (Russian). Otsenka parametrov. Proverka gipotez. [Estimation of parameters. Testing of hypotheses]. MR**782295****2.**V. N. Vapnik and A. Ya. Chervonenkis,*Teoriya raspoznavaniya obrazov. Statisticheskie problemy obucheniya*, Izdat. “Nauka”, Moscow, 1974 (Russian). MR**0474638****3.**O. Kubaychuk,*Estimation of moments by observations from mixtures with varying concentrations*, Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko (Kyiv, 2002), 2002, pp. 226–231. MR**2027394****4.**O. O. Kubaychuk,*Estimation of moments from mixtures with the use of improved weighted empirical distribution function*, Visnyk Kyiv Univ. Matematyka. Mekhanika**9-10**(2003), 48-52. (Ukrainian)**5.**O. O. Kubaychuk,*Statistical analysis of characteristics of random variables with the use of observations of a mixture*, Ph.D. Dissertation, Kyiv National University, Kyiv, 2004. (Ukrainian)**6.**N. Lodatko and R. Maĭboroda,*Estimation of the density of a distribution from observations with an admixture*, Teor. Ĭmovīr. Mat. Stat.**73**(2005), 88–96 (Ukrainian, with Russian summary); English transl., Theory Probab. Math. Statist.**73**(2006), 99–108. MR**2213844**, 10.1090/S0094-9000-07-00684-9**7.**R. E. Maĭboroda,*Statistical Analysis of Mixtures. A Course of Lectures*, Kyiv National University, Kyiv, 2003. (Ukrainian)**8.**P. J. Bickel,*On adaptive estimation*, Ann. Statist.**10**(1982), no. 3, 647–671. MR**663424****9.**Xiaotong Shen,*On methods of sieves and penalization*, Ann. Statist.**25**(1997), no. 6, 2555–2591. MR**1604416**, 10.1214/aos/1030741085**10.**Charles J. Stone,*Adaptive maximum likelihood estimators of a location parameter*, Ann. Statist.**3**(1975), 267–284. MR**0362669**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2000):
62G07,
62G20

Retrieve articles in all journals with MSC (2000): 62G07, 62G20

Additional Information

**N. Lodatko**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, 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 Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine

Email:
mre@univ.kiev.ua

DOI:
http://dx.doi.org/10.1090/S0094-9000-08-00715-1

Keywords:
Method of moments,
adaptive estimator,
a mixture with varying concentrations,
consistency,
asymptotic normality,
asymptotic variance

Received by editor(s):
September 19, 2005

Published electronically:
January 24, 2008

Article copyright:
© Copyright 2008
American Mathematical Society