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].

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 |