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Theory of Probability and Mathematical Statistics

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Statistical analysis of curve fitting methods in errors-in-variables models

Authors: A. Al-Sharadqah and N. Chernov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 84 (2011).
Journal: Theor. Probability and Math. Statist. 84 (2012), 1-14
MSC (2010): Primary 62H10, 62J02; Secondary 62H35
Published electronically: July 26, 2012
MathSciNet review: 2857412
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Abstract: Regression models in which all variables are subject to errors are known as errors-in-variables (EIV) models. The respective parameter estimates have many unusual properties: their exact distributions are very hard to determine, and their absolute moments are often infinite (so that their mean and variance do not exist). In our paper, Error analysis for circle fitting algorithms, Electr. J. Stat. 3 (2009), 886-911, we developed an unconventional statistical analysis that allowed us to effectively assess EIV parameter estimates and design new methods with superior characteristics. In this paper we validate our approach in a series of numerical tests. We also prove that in the case of fitting circles, the estimates of the parameters are absolutely continuous (have densities).

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  • 1. R. J. Adcock, Note on the method of least squares, Analyst 4 (1877), 183-184.
  • 2. A. Al-Sharadqah and N. Chernov, Error analysis for circle fitting algorithms, Electr. J. Stat. 3 (2009), 886-911. MR 2540845 (2010j:62170)
  • 3. Y. Amemiya and W. A. Fuller, Estimation for the nonlinear functional relationship, Annals Statist. 16 (1988), 147-160. MR 924862 (90c:62065)
  • 4. T. W. Anderson, Estimation of linear functional relationships: Approximate distributions and connections with simultaneous equations in econometrics, J. R. Statist. Soc. B 38 (1976), 1-36. MR 0411025 (53:14764)
  • 5. T. W. Anderson and T. Sawa, Exact and approximate distributions of the maximum likelihood estimator of a slope coefficient, J. R. Statist. Soc. B 44 (1982), 52-62. MR 655374 (84j:62023)
  • 6. M. Berman, Large sample bias in least squares estimators of a circular arc center and its radius, CVGIP: Image Understanding 45 (1989), 126-128.
  • 7. A. W. Bowman and A. Azzalini, Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations, Oxford Statistical Science Series, vol. 18, Oxford University Press, 1997.
  • 8. R. J. Carroll, D. Ruppert, L. A. Stefansky, and C. M. Crainiceanu, Measurement Error in Nonlinear Models: A Modern Perspective, Chapman & Hall, London, 2006. MR 2243417 (2007e:62004)
  • 9. N. N. Chan, On circular functional relationships, J. R. Statist. Soc. B 27 (1965), 45-56. MR 0189163 (32:6590)
  • 10. C.-L. Cheng and A. Kukush, Non-existence of the first moment of the adjusted least squares estimator in multivariate errors-in-variables model, Metrika 64 (2006), 41-46. MR 2242556 (2008b:62042)
  • 11. C.-L. Cheng and J. W. Van Ness, On estimating linear relationships when both variables are subject to errors, J. R. Statist. Soc. B 56 (1994), 167-183. MR 1257805
  • 12. C.-L. Cheng and J. W. Van Ness, Statistical Regression with Measurement Error, Arnold, London, 1999. MR 1719513 (2001k:62001)
  • 13. N. Chernov, Fitting circles to scattered data: parameter estimates have no moments, Metrika 73 (2011), 373-384. MR 2785031 (2012h:62252)
  • 14. N. Chernov, Circular and Linear Regression: Fitting Circles and Lines by Least Squares, Monographs on Statistics & Applied Probability, vol. 117, CRC Press, Boca Raton-London-New York, 2010. MR 2723019 (2012a:62005)
  • 15. N. Chernov and C. Lesort, Statistical efficiency of curve fitting algorithms, Comp. Stat. Data Anal. 47 (2004), 713-728. MR 2101548 (2005f:62038)
  • 16. N. Chernov and C. Lesort, Least squares fitting of circles, J. Math. Imag. Vision 23 (2005), 239-251. MR 2181705 (2007d:68165)
  • 17. L. J. Gleser, Functional, structural and ultrastructural errors-in-variables models, Proc. Bus. Econ. Statist. Sect. Am. Statist. Ass., 1983, pp. 57-66.
  • 18. R. Z. Hasminskii and I. A. Ibragimov, On Asymptotic Efficiency in the Presence of an Infinite-dimensional Nuisance Parameter, Lecture Notes in Math, vol. 1021, Springer, Berlin, 1983, pp. 195-229. MR 735986 (85h:62040)
  • 19. S. van Huffel (ed.), Total Least Squares and Errors-in-Variables Modeling, Kluwer, Dordrecht, 2002. MR 1951009 (2003g:00026)
  • 20. J. P. Imhof, Computing the distribution of quadratic forms in normal variables, Biometrika 4 (1961), 419-426. MR 0137199 (25:655)
  • 21. K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice, Elsevier, Amsterdam, 1996. MR 1392697 (97k:62133)
  • 22. K. Kanatani, Cramer-Rao lower bounds for curve fitting, Graph. Mod. Image Process. 60 (1998), 93-99.
  • 23. K. Kanatani, For geometric inference from images, what kind of statistical model is necessary?, Syst. Comp. Japan 35 (2004), 1-9.
  • 24. K. Kanatani, Optimality of maximum likelihood estimation for geometric fitting and the KCR lower bound, Memoirs Fac. Engin. Okayama Univ. 39 (2005), 63-70.
  • 25. K. Kanatani, Statistical optimization for geometric fitting: Theoretical accuracy bound and high order error analysis, Int. J. Computer Vision 80 (2008), 167-188.
  • 26. A. Kukush and E.-O. Maschke, The efficiency of adjusted least squares in the linear functional relationship, J. Multivar. Anal. 87 (2003), 261-274. MR 2016938 (2004m:62155)
  • 27. A. M. Mathai and S. B. Provost, Quadratic Forms in Random Variables, Marcel Dekker, New York, 1992. MR 1192786 (94g:62110)
  • 28. Y. Nievergelt, A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres, J. Numerische Math. 91 (2002), 257-303. MR 1900920 (2003d:52011)
  • 29. J. Robinson, The distribution of a general quadratic form in normal variables, Austral. J. Statist. 7 (1965), 110-114. MR 0198584 (33:6739)
  • 30. P. Rangarajan and K. Kanatani, Improved algebraic methods for circle fitting, Electr. J. Statist. 3 (2009), 1075-1082. MR 2557129 (2011b:68220)
  • 31. E. Zelniker and V. Clarkson, A statistical analysis of the Delogne-Kåsa method for fitting circles, Digital Signal Proc. 16 (2006), 498-522.

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Additional Information

A. Al-Sharadqah
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294

N. Chernov
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294

Keywords: Errors-in-variables, regression, curve fitting, circle fitting, line fitting, functional model
Received by editor(s): March 4, 2010
Published electronically: July 26, 2012
Additional Notes: The second author was partially supported by National Science Foundation, grant DMS-0652896
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society