Brazilian Journal of Probability and Statistics

Errors-In-Variables regression and the problem of moments

Ali Al-Sharadqah, Nikolai Chernov, and Qizhuo Huang

Full-text: Open access


In regression problems where covariates are subject to errors (albeit small) it often happens that maximum likelihood estimators (MLE) of relevant parameters have infinite moments. We study here circular and elliptic regression, that is, the problem of fitting circles and ellipses to observed points whose both coordinates are measured with errors. We prove that several popular circle fits due to Pratt, Taubin, and others return estimates of the center and radius that have infinite moments. We also argue that estimators of the ellipse parameters (center and semiaxes) should have infinite moments, too.

Article information

Braz. J. Probab. Stat., Volume 27, Number 4 (2013), 401-415.

First available in Project Euclid: 9 September 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Errors-In-Variables regression moments fitting circles fitting ellipses


Al-Sharadqah, Ali; Chernov, Nikolai; Huang, Qizhuo. Errors-In-Variables regression and the problem of moments. Braz. J. Probab. Stat. 27 (2013), no. 4, 401--415. doi:10.1214/11-BJPS173.

Export citation


  • Adcock, R. (1877). Note on the method of least squares. Analyst London 4, 183–184.
  • Ahn, S. J. (2004). Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space. LNCS 3151. Berlin: Springer.
  • Al-Sharadqah, A. and Chernov, N. (2009). Error analysis for circle fitting algorithms. Electronic Journal of Statistics 3, 886–911.
  • Al-Sharadqah, A. and Chernov, N. (2011). Statistical analysis of curve fitting methods in Errors-In-Variables models. Theor. Imovir. ta Matem. Statyst. 84, 4–17.
  • Anderson, T. W. (1976). Estimation of linear functional relationships: Approximate distributions and connections with simultaneous equations in econometrics. Journal of the Royal Statistical Society, Ser. B 38, 1–36.
  • Anderson, T. W. and Sawa, T. (1982). Exact and approximate distributions of the maximum likelihood estimator of a slope coefficient. Journal of the Royal Statistical Society, Ser. B 44, 52–62.
  • Chan, N. N. (1965). On circular functional relationships. Journal of the Royal Statistical Society, Ser. B 27, 45–56.
  • Cheng, C. L. and Kukush, A. (2006). Non-existence of the moments of the adjusted least squares estimator in multivariate Errors-In-Variables model. Metrika 64, 41–46.
  • Chernov, N. I. and Ososkov, G. A. (1984). Effective algorithms for circle fitting. Computer Physics Communications 33, 329–333.
  • Chernov N. (2010). Circular and Linear Regression: Fitting Circles and Lines by Least Squares. Boca Raton: Chapman & Hall.
  • Chernov, N. (2011). Fitting circles to scattered data: Parameter estimates have no moments. Metrika 73, 373–384.
  • Kanatani, K. (2004). For geometric inference from images, what kind of statistical model is necessary? Systems and Computers in Japan 35, 1–9.
  • Kåsa, I. (1976). A curve fitting procedure and its error analysis. IEEE Trans. Inst. Meas. 25, 8–14.
  • Kukush, A., Markovsky, I. and Van Huffel, S. (2004). Consistent estimation in an implicit quadratic measurement error model. Computational Statistics Data Analysis 47, 123–147.
  • Nievergelt, Y. (2004). Fitting conics of specific types to data. Linear Algebra and Its Applications 378, 1–30.
  • Pratt, V. (1987). Direct least-squares fitting of algebraic surfaces. Computer Graphics 21, 145–152.
  • Taubin, G. (1991). Estimation of planar curves, surfaces and nonplanar space curves defined by implicit equations, with applications to edge and range image segmentation. IEEE Trans. Pattern Analysis Machine Intelligence 13, 1115–1138.
  • Zelniker, E. and Clarkson, V. (2006). A statistical analysis of the Delogne–Kåsa method for fitting circles. Digital Signal Processing 16, 498–522.