Singular asymptotic normality of an estimator in the conic section fitting problem. II
Author:
S. V. Shklyar
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 93 (2016), 177-196
MSC (2010):
Primary 65D10; Secondary 62F12
DOI:
https://doi.org/10.1090/tpms/1002
Published electronically:
February 7, 2017
MathSciNet review:
3553449
Full-text PDF Free Access
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Additional Information
Abstract: This is the second part of the author paper published in Theor. Probability and Math. Statist. 92 (2016), 147–161. The first part considers the functional version of the conic section fitting problem and states the asymptotic normality of the ALS2 estimator for the coefficients of the conic section. In the present paper, a similar theorem on the asymptotic normality is obtained for the structural model. Two estimators of the asymptotic covariance matrix are constructed. The consistency of both estimators is proved.
References
- Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu, Measurement error in nonlinear models, 2nd ed., Monographs on Statistics and Applied Probability, vol. 105, Chapman & Hall/CRC, Boca Raton, FL, 2006. A modern perspective. MR 2243417, DOI 10.1201/9781420010138
- G. M. Fichtenholz, A Course of Differential and Integral Calculus, vol. 1, “Fizmatlit”, Moscow, 2003. (Russian)
- Alexander Kukush, Ivan Markovsky, and Sabine Van Huffel, Consistent estimation in an implicit quadratic measurement error model, Comput. Statist. Data Anal. 47 (2004), no. 1, 123–147. MR 2087933, DOI 10.1016/j.csda.2003.10.022
- Valentin V. Petrov, Limit theorems of probability theory, Oxford Studies in Probability, vol. 4, The Clarendon Press, Oxford University Press, New York, 1995. Sequences of independent random variables; Oxford Science Publications. MR 1353441
- Steven Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. MR 741185
- Sergiy Shklyar, Alexander Kukush, Ivan Markovsky, and Sabine Van Huffel, On the conic section fitting problem, J. Multivariate Anal. 98 (2007), no. 3, 588–624. MR 2293016, DOI 10.1016/j.jmva.2005.12.003
- S. V. Shklyar, Singular asymptotic normality of an estimator in the conic section fitting problem. I, Teor. Ĭmov$\bar {\text {\i }}$r. Mat. Stat. 92 (2015), 137–150; English transl. in Theor. Probability and Math. Statist. 92 (2016), 147–161.
- S. V. Shklyar, ALS2 estimator in two lines fitting model, International Conference “Stochastic Processes in Abstract Spaces”, Institute of Mathematics, Kyiv, 2015, p. 61.
References
- R. J. Carroll, D. Ruppert, L. A. Stefanski, and C. M. Crainiceanu, Measurement Error in Nonlinear Models: A Modern Perspective, Chapman & Hall, Boca Raton, 2006. MR 2243417
- G. M. Fichtenholz, A Course of Differential and Integral Calculus, vol. 1, “Fizmatlit”, Moscow, 2003. (Russian)
- A. Kukush, I. Markovsky, and S. Van Huffel, Consistent estimation in an implicit quadratic measurement error model, Comput. Statist. Data Anal. 47 (2004), no. 1, 123–147. MR 2087933
- V. V. Petrov, Limit Theorems of Probability Theory. Sequences of Independent Random Variables, “Nauka”, Moscow, 1987; English transl., The Clarendon Press, Oxford University Press, New York, 1995. MR 1353441
- S. Roman, The Umbral Calculus, Academic Press, Orlando, 1984. MR 741185
- S. Shklyar, A. Kukush, I. Markovsky, and S. Van Huffel, On the conic section fitting problem, J. Multivariate Anal. 98 (2007), no. 3, 588–624. MR 2293016
- S. V. Shklyar, Singular asymptotic normality of an estimator in the conic section fitting problem. I, Teor. Ĭmov$\bar {\text {\i }}$r. Mat. Stat. 92 (2015), 137–150; English transl. in Theor. Probability and Math. Statist. 92 (2016), 147–161.
- S. V. Shklyar, ALS2 estimator in two lines fitting model, International Conference “Stochastic Processes in Abstract Spaces”, Institute of Mathematics, Kyiv, 2015, p. 61.
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Additional Information
S. V. Shklyar
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
shklyar@mail.univ.kiev.ua
Keywords:
Errors in variables,
asymptotic normality,
estimation of parameters of a conic section,
estimation of the asymptotic variance
Received by editor(s):
September 7, 2015
Published electronically:
February 7, 2017
Additional Notes:
This paper was prepared following the talk at the International conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015
Article copyright:
© Copyright 2017
American Mathematical Society
