On Consistent Estimators in Linear and Bilinear Multivariate Errors-In-Variables Models

Alexander Kukush (), Ivan Markovsky () and Sabine Van Huffel ()
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Alexander Kukush: ESAT-SISTA, K. U.Leuven
Ivan Markovsky: ESAT-SISTA, K. U.Leuven
Sabine Van Huffel: ESAT-SISTA, K. U.Leuven

A chapter in Total Least Squares and Errors-in-Variables Modeling, 2002, pp 155-164 from Springer

Abstract: Abstract We consider three multivariate regression models related to the TLS problem. The errors are allowed to have unequal variances. For the model AX = B, the elementwise-weighted TLS estimator is considered. The matrix [A B] is observed with errors and has independent rows, but the errors in a row are correlated. In addition, the corresponding error covariance matrices may differ from row to row and some of the columns are allowed to be error-free. We give mild conditions for weak consistency of the estimator, when the number of rows in A increases. We derive the objective function for the estimator and propose an iterative procedure to compute the solution. In a bilinear model AXB = C, where the data A, B, C are perturbed by errors, an adjusted least squares estimator is considered, which is consistent, i.e. converges to X, as the number m of rows in A and the number q of columns in B increase. A similar approach is applied in a related model, arising in motion analysis. The model is v T Fu = 0, where the vectors u and v are homogeneous coordinates of the projections of the same rigid object point in two images, and F is a rank deficient matrix. Each pair (u, v) is observed with measurement errors. We construct a consistent estimator of F in three steps: a) estimate the measurement error variance, b) construct a preliminary matrix estimate, and c) project that estimate on the subspace of singular matrices. A simulation study illustrates the theoretical results.

Keywords: linear and bilinear errors-in-variables models; elementwise-weighted total least squares; adjusted least squares; essential/fundamental matrix. (search for similar items in EconPapers)
Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-017-3552-0_14

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DOI: 10.1007/978-94-017-3552-0_14

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