Equivariant adjusted least squares estimator in two-line fitting model. (English) Zbl 1349.62318
Summary: 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.
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.
MSC:
| 62J05 | Linear regression; mixed models |
| 62H12 | Estimation in multivariate analysis |
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
References:
| [1] | Ahn, S. J., Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space (2004) · Zbl 1067.65019 · doi:10.1007/b104017 |
| [2] | Bilmes, J.A.: A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models. Technical report TR-97-021, International Computer Science Institute (1998) |
| [3] | Cheng, C.-L.; Van Ness, J. W., Statistical Regression with Measurement Error (1999) · Zbl 0947.62046 |
| [4] | Chiang, C. L., On regular best asymptotically normal estimates, Ann. Math. Stat., 27, 2, 336-351 (1956) · Zbl 0074.35205 · doi:10.1214/aoms/1177728262 |
| [5] | Fazekas, I.; Kukush, A.; Zwanzig, S., Correction of nonlinear orthogonal regression estimator, Ukr. Math. J., 56, 8, 1308-1330 (2004) · Zbl 1070.62053 · doi:10.1007/s11253-005-0059-0 |
| [6] | 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) · Zbl 1011.62069 · doi:10.1016/S0167-9473(02)00068-3 |
| [7] | Kukush, A.; Markovsky, I.; Van Huffel, S., Correction of nonlinear orthogonal regression estimator, Comput. Stat. Data Anal., 47, 1, 123-147 (2004) · Zbl 1429.62295 · doi:10.1016/j.csda.2003.10.022 |
| [8] | 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) · Zbl 1164.93014 · doi:10.1002/nla.361 |
| [9] | Repetatska, G., An improved orthogonal regression estimator for the implicit functional errors-in-variables model, Bull. Kyiv Natl. Taras Shevchenko Univ. Math. Mech., 23, 37-45 (2010) · Zbl 1223.62112 |
| [10] | Rohatgi, V. K.; Székely, G. J., Sharp inequalities between skewness and kurtosis, Stat. Probab. Lett., 8, 4, 296-299 (1989) · Zbl 0682.60014 · doi:10.1016/0167-7152(89)90035-7 |
| [11] | Shklyar, S.; Kukush, A.; Markovsky, I.; Van Huffel, S., On the conic section fitting problem, J. Multivar. Anal., 98, 3, 588-624 (2007) · Zbl 1118.62062 · doi:10.1016/j.jmva.2005.12.003 |
| [12] | Shklyar, S. V., Singular asymptotic normality of an estimator in the conic section fitting problem. I, Teor. Imovir. Mat. Stat., 92, 137-150 (2015) · Zbl 1343.62010 |
| [13] | Shklyar, S. V., Singular asymptotic normality of an estimator in the conic section fitting problem. II, Teor. Imovir. Mat. Stat., 93, 163-180 (2015) · Zbl 1357.65019 |
| [14] | 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 |
| [15] | Waibel, P.; Matthes, J.; Gröll, L., Constrained ellipse fitting with center on a line, J. Math. Imaging Vis., 53, 3, 364-382 (2015) · Zbl 1343.68256 · doi:10.1007/s10851-015-0584-x |
| [16] | Zelnik-Manor, L.; Irani, M., Proceedings of the 7th IEEE International Conference on Computer Vision, 1999, 2, 710-7152 (1999) · doi:10.1109/ICCV.1999.790291 |
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