Equivariant adjusted least squares estimator in two-line fitting model
Volume 3, Issue 1 (2016), pp. 19–45
Pub. online: 21 March 2016
Type: Research Article
Open Access
Open Access
Received
30 January 2016
30 January 2016
Revised
19 February 2016
19 February 2016
Accepted
19 February 2016
19 February 2016
Published
21 March 2016
21 March 2016
Abstract
We consider the two-line fitting problem. True points lie on two straight lines and are observed with Gaussian perturbations. For each observed point, it is not known on which line the corresponding true point lies. The parameters of the lines are estimated.
This model is a restriction of the conic section fitting model because a couple of two lines is a degenerate conic section. The following estimators are constructed: two projections of the adjusted least squares estimator in the conic section fitting model, orthogonal regression estimator, parametric maximum likelihood estimator in the Gaussian model, and regular best asymptotically normal moment estimator.
The conditions for the consistency and asymptotic normality of the projections of the adjusted least squares estimator are provided. All the estimators constructed in the paper are equivariant. The estimators are compared numerically.
References
Ahn, S.J.: Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space. Springer, Heidelberg (2004). doi:10.1007/b104017
Cheng, C.-L., Van Ness, J.W.: Statistical Regression with Measurement Error. Arnold, London (1999). MR1719513
Chiang, C.L.: On regular best asymptotically normal estimates. Ann. Math. Stat. 27(2), 336–351 (1956). doi:10.1214/aoms/1177728262 MR0089558
Fazekas, I., Kukush, A., Zwanzig, S.: Correction of nonlinear orthogonal regression estimator. Ukr. Math. J. 56(8), 1308–1330 (2004). MR2136312. doi:10.1007/s11253-005-0059-0
Kukush, A., Markovsky, I., Van Huffel, S.: Consistent fundamental matrix estimation in a quadratic measurement error model arising in motion analysis. Comput. Stat. Data Anal. 41(1), 3–18 (2002). MR1944689. doi:10.1016/S0167-9473(02)00068-3
Kukush, A., Markovsky, I., Van Huffel, S.: Correction of nonlinear orthogonal regression estimator. Comput. Stat. Data Anal. 47(1), 123–147 (2004). MR2087933. doi:10.1016/j.csda.2003.10.022
Markovsky, I., Van Huffel, S., Kukush, A.: On the computation of the multivariate structured total least squares estimator. Numer. Linear Algebra Appl. 11(5–6), 591–608 (2004). MR2067822. doi:10.1002/nla.361
Rohatgi, V.K., Székely, G.J.: Sharp inequalities between skewness and kurtosis. Stat. Probab. Lett. 8(4), 296–299 (1989). MR1028986. doi:10.1016/0167-7152(89)90035-7
Shklyar, S., Kukush, A., Markovsky, I., Van Huffel, S.: On the conic section fitting problem. J. Multivar. Anal. 98(3), 588–624 (2007). MR2293016. doi:10.1016/j.jmva.2005.12.003
Vidal, R., Ma, Y., Sastry, S.: Generalized principal component analysis (GPCA). IEEE Trans. Pattern Anal. Mach. Intell. 27(12), 1945–1959 (2005). doi:10.1109/TPAMI.2005.244
Waibel, P., Matthes, J., Gröll, L.: Constrained ellipse fitting with center on a line. J. Math. Imaging Vis. 53(3), 364–382 (2015). MR3397105. doi:10.1007/s10851-015-0584-x
Zelnik-Manor, L., Irani, M.: Multi-view subspace constraints on homographies. In: Proceedings of the 7th IEEE International Conference on Computer Vision, 1999, vol. 2, pp. 710–7152 (1999). doi:10.1109/ICCV.1999.790291
