An estimator of the location parameter obtained from observations with admixture
Author:
O. Sugakova
Translated by:
N. N. Semenov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 80 (2009).
Journal:
Theor. Probability and Math. Statist. 80 (2010), 143152
MSC (2000):
Primary 62G07; Secondary 62G20
Published electronically:
August 20, 2010
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider a model of observations from a twocomponent mixture such that the distribution of the admixture is known, while the distribution of the other component (treated as the primary one) is unknown. We assume that the distribution of the primary component is symmetric about the location parameter. We propose a method for constructing an unbiased estimating equation for the location parameter of the primary component. The asymptotic normality of the corresponding estimators is proved. The exact lower bound for the asymptotic variance is found and estimating functions for which this bound is attained are described. It is shown that the exact lower bound for the estimators under consideration is close to the corresponding bound of effectiveness of parametric estimators for the case where the distributions of both components are Gaussian.
 1.
N.
Lodatko and R.
Maĭboroda, An adaptive moment estimate for the
distribution parameter from observations with an admixture, Teor.
Ĭmovīr. Mat. Stat. 75 (2006), 61–70
(Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 75 (2007), 71–82. MR 2321182
(2008g:62101), http://dx.doi.org/10.1090/S0094900008007151
 2.
A.
A. Borovkov, Matematicheskaya statistika, “Nauka”,
Moscow, 1984 (Russian). Otsenka parametrov. Proverka gipotez. [Estimation
of parameters. Testing of hypotheses]. MR 782295
(86i:62001)
A.
A. Borovkov, Mathematical statistics, Gordon and Breach
Science Publishers, Amsterdam, 1998. Translated from the Russian by A.
Moullagaliev and revised by the author. MR 1712750
(2000f:62003)
 3.
Laurent
Bordes, Stéphane
Mottelet, and Pierre
Vandekerkhove, Semiparametric estimation of a twocomponent mixture
model, Ann. Statist. 34 (2006), no. 3,
1204–1232. MR 2278356
(2008e:62064), http://dx.doi.org/10.1214/009053606000000353
 4.
Laurent
Bordes, Céline
Delmas, and Pierre
Vandekerkhove, Semiparametric estimation of a twocomponent mixture
model where one component is known, Scand. J. Statist.
33 (2006), no. 4, 733–752. MR 2300913
(2008f:62049), http://dx.doi.org/10.1111/j.14679469.2006.00515.x
 5.
David
R. Hunter, Shaoli
Wang, and Thomas
P. Hettmansperger, Inference for mixtures of symmetric
distributions, Ann. Statist. 35 (2007), no. 1,
224–251. MR 2332275
(2008g:62079), http://dx.doi.org/10.1214/009053606000001118
 6.
Geoffrey
McLachlan and David
Peel, Finite mixture models, Wiley Series in Probability and
Statistics: Applied Probability and Statistics, WileyInterscience, New
York, 2000. MR
1789474 (2002b:62025)
 7.
Simon
Newcomb, A Generalized Theory of the Combination of Observations so
as to Obtain the Best Result, Amer. J. Math. 8
(1886), no. 4, 343–366. MR
1505430, http://dx.doi.org/10.2307/2369392
 8.
K. Pearson, Contribution to the mathematical theory of evolution, Phil. Trans. Roy. Soc. A 185 (1894), 71110.
 9.
Jun
Shao, Mathematical statistics, 2nd ed., Springer Texts in
Statistics, SpringerVerlag, New York, 2003. MR 2002723
(2004g:62002)
 10.
D.
M. Titterington, A.
F. M. Smith, and U.
E. Makov, Statistical analysis of finite mixture
distributions, Wiley Series in Probability and Mathematical
Statistics: Applied Probability and Statistics, John Wiley & Sons,
Ltd., Chichester, 1985. MR 838090
(87j:62033)
 1.
 N. Lodatko and R. Maĭboroda, An adaptive moment estimator of a parameter of a distribution constructed from observations with admixture, Teor. Imovirnost. Matem. Statyst. 75 (2006), 6170; English transl. in Theor. Probability and Math. Statist. 75 (2007), 7182. MR 2321182 (2008g:62101)
 2.
 A. A. Borovkov, Mathematical Statistics, Nauka, Moscow, 1984; English transl., Gordon and Breach Science Publishers, Amsterdam, 1998. MR 782295 (86i:62001); MR 1712750 (2000f:62003)
 3.
 L. Bordes, S. Mottelet, and P. Vandekerkhove, Semiparametric estimation of a twocomponent mixture model, Ann. Statist. 34 (2006), no. 3, 12041232. MR 2278356 (2008e:62064)
 4.
 L. Bordes, C. Delmas, and P. Vandekerkhove, Semiparametric estimation of a twocomponent mixture model where one component is known, Scand. J. Statist. 33 (2006), 733752. MR 2300913 (2008f:62049)
 5.
 D. R. Hunter, S. Wang, and T. R. Hettmansperger, Inference for mixtures of symmetric distributions, Ann. Statist. 35 (2007), 224251. MR 2332275 (2008g:62079)
 6.
 G. J. McLachlan and D. Peel, Finite Mixture Models, Wiley, New York, 2000. MR 1789474 (2002b:62025)
 7.
 S. Newcomb, A generalized theory of the combination of observations so as to obtain the best result, Amer. J. Math. 8 (1886), 343366. MR 1505430
 8.
 K. Pearson, Contribution to the mathematical theory of evolution, Phil. Trans. Roy. Soc. A 185 (1894), 71110.
 9.
 J. Shao, Mathematical Statistics, 2nd ed., SpringerVerlag, New York, 2003. MR 2002723 (2004g:62002)
 10.
 D. M. Titterington, A. F. Smith, and U. E. Makov, Analysis of Finite Mixture Distributions, Wiley, New York, 1985. MR 0838090 (87j:62033)
Similar Articles
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
O. Sugakova
Affiliation:
Department of Higher Mathematics and Theoretical Radiophysics, Faculty for Radiophysics, National Taras Shevchenko University, Academician Glushkov Avenue, 2, Kiev 03127, Ukraine
Email:
sugak@univ.kiev.ua
DOI:
http://dx.doi.org/10.1090/S009490002010008015
PII:
S 00949000(2010)008015
Keywords:
Method of moments,
a finite mixture of probability distributions,
consistency,
asymptotic normality,
asymptotic variance
Received by editor(s):
February 25, 2008
Published electronically:
August 20, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
