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On consistent estimators in linear and bilinear multivariate errors-in-variables models. (English) Zbl 0995.65015
Van Huffel, Sabine (ed.) et al., Total least squares and errors-in-variables modeling. Analysis, algorithms and applications. Dordrecht: Kluwer Academic Publishers. 155-164 (2002).
Summary: We consider three multivariate regression models related to the total least squares (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.
For the entire collection see [Zbl 0984.00011].

##### MSC:
 65C60 Computational problems in statistics (MSC2010) 62J12 Generalized linear models (logistic models) 62H12 Estimation in multivariate analysis