Orthogonal regression method for observations from a mixture

Authors:
R. E. Maĭboroda, G. V. Navara and O. V. Sugakova

Translated by:
S. V. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **99** (2018).

Journal:
Theor. Probability and Math. Statist. **99** (2019), 169-188

MSC (2010):
Primary 62G05, 62G20; Secondary 62J05

DOI:
https://doi.org/10.1090/tpms/1088

Published electronically:
February 27, 2020

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A generalization of the orthogonal regression method is considered for estimating parameters of the simple linear regression model with errors in variables for observations from a mixture with varying concentrations. The consistency and asymptotic normality is proved for the estimators studied in the paper. The dispersion matrix is established.

**[1]**C.-L. Cheng and J. Van Ness,*Statistical Regression with Measurement Error*, Kendall's Library of Statistics 6, Arnold, London, 1999. MR**1719513****[2]**R. Maĭboroda and O. Sugakova,*Estimation and Classification by Using Observations from a Mixture*, ``Kyiv University'', Kyiv, 2008. (in Ukrainian)**[3]**S. Masiuk, A. Kukush, S. Shklyar, M. Chepurny, and I. Likhtarov (eds.),*Radiation Risk Estimation: Based on Measurement Error Models*, 2nd ed., de Gruyter series in Mathematics and Life Sciences, vol. 5, de Gruyter, 2017. MR**3726857****[4]**T. Benaglia, D. Chauveau, D. Hunter, and D. Young,, J. Stat. Software`mixtools`: An R Package for Analyzing Finite Mixture Models**32**(2009), no. 6, 1-29.**[5]**S. Faria and G. Soromenhob,*Fitting mixtures of linear regressions*, J. Stat. Computation and Simulation**80**(2010), no. 2, 201-225. MR**2757044****[6]**B. Grün and F. Leisch,*Fitting finite mixtures of linear regression models with varying & fixed effects in R*, Proceedings in Computational Statistics, Physica Verlag, Heidelberg, Germany, 2006, pp. 853-860. MR**2370869****[7]**R. Maiboroda and O. Sugakova,*Statistics of mixtures with varying concentrations with application to DNA microarray data analysis*, J. Nonparam. Stat.**24**(2012), no. 1, 201-205. MR**2885834****[8]**D. Liubashenko and R. Maiboroda,*Linear regression by observations from mixtures with varying concentrations*, Modern Stochastics: Theory and Applications**2**(2015), 343-353. MR**3456142****[9]**J. Shao,*Mathematical Statistics*, Springer-Verlag, New York, 1998. MR**1670883****[10]**R. Branham,*Total Least Squares in Astronomy*, Total Least Squares and Errors-in-Variables Modeling, Springer, Dordrecht, 2002, pp. 375-384. MR**1952962**

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Additional Information

**R. E. Maĭboroda**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine

Email:
mre@univ.kiev.ua

**G. V. Navara**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine

Email:
mrswade111017@gmail.com

**O. V. Sugakova**

Affiliation:
Department of Mathematics and Theoretical Radiophysics, Faculty for Radiophysics, Electronics, and Computer Systems, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine

Email:
sugak@univ.kiev.ua

DOI:
https://doi.org/10.1090/tpms/1088

Keywords:
Model of a mixture,
orthogonal regression,
method of generalized estimating equations

Received by editor(s):
March 4, 2018

Published electronically:
February 27, 2020

Article copyright:
© Copyright 2020
American Mathematical Society