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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
DOI: https://doi.org/10.1090/tpms/989
Published electronically: August 10, 2016
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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.


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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
Email: shklyar@mail.univ.kiev.ua

DOI: https://doi.org/10.1090/tpms/989
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