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Singular asymptotic normality of an estimator in the conic section fitting problem. I

Author: S. V. Shklyar
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 92 (2015).
Journal: Theor. Probability and Math. Statist. 92 (2016), 147-161
MSC (2010): Primary 65D10; Secondary 62F12
Published electronically: August 10, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: The conic section fitting problem is considered. True points are assumed to lie on a conic section. The points are observed with additive errors, which are independent and have bivariate normal distribution $ N(0, \sigma ^2 I)$ with unknown $ \sigma ^2$. We study asymptotic properties of the estimator of conic section parameters introduced by Kukush, Markovsky, and Van Huffel in Computational Statistics and Data Analysis 47 (2004), 123-147. Sufficient conditions for singular asymptotic normality of the estimator are provided. The asymptotic covariance matrix is singular and has defect 1 because the unit sphere in Euclidean space is taken as a parameter space.

References [Enhancements On Off] (What's this?)

  • 1. R. J. Carroll, D. Ruppert, L. A. Stefanski, and C. Crainiceanu, Measurement Error in Nonlinear Models: A Modern Perspective, second edition, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2243417
  • 2. A. Kukush, I. Markovsky, and S. Van Huffel, Consistent estimation in an implicit quadratic error model, Comput. Statist. Data Anal. 47 (2004), no. 1, 123-147. MR 2087933
  • 3. A. Kukush and H. Schneeweiss, Comparing different estimators in a nonlinear measurement error model. I, Math. Methods Statist. 14 (2005), no. 1, 53-79. MR 2158071
  • 4. I. Markovsky, A. Kukush, and S. Van Huffel, Consistent least squares fitting of ellipsoids, Numer. Math. 98 (2004), no. 1, 177-194. MR 2076059
  • 5. J. Pfanzagl, On the measurability and consistency of minimum contrast estimates, Metrika 14 (1969), 249-272.
  • 6. 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
  • 7. G. W. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, Boston, 1990. MR 1061154
  • 8. P. K. Suetin, Orthogonal Polynomials in Two Variables, ``Nauka'', Moscow, 1988; English transl. Gordon and Breach Science Publishers, Amsterdam, 1999. MR 978197

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Additional Information

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

Keywords: Errors in variables, asymptotic normality, estimation of parameters of a conic section
Received by editor(s): December 23, 2014
Published electronically: August 10, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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