Consistency of elementwise-weighted total least squares estimator in a multivariate errors-in-variables model <Emphasis Type="Bold">AX=B</Emphasis>

A multivariate measurement error model AXB is considered. The errors in A,B] are rowwise independent, but within each row the errors may be correlated. Some of the columns are observed without errors, and in addition the error covariance matrices may differ from row to row. The total covariance structure of the errors is supposed to be known up to a scalar factor. The fully weighted total least squares estimator of X is studied, which in the case of normal errors coincides with the maximum likelihood estimator. We give mild conditions for weak and strong consistency of the estimator, when the number of rows in A increases. The results generalize the conditions of Gallo given for a univariate homoscedastic model (where B is a vector), and extend the conditions of Gleser given for the multivariate homoscedastic model. We derive the objective function for the estimator and propose an iteratively reweighted numerical procedure.Acknowledgements.A. Kukush is supported by a postdoctoral research fellowship of the Belgian office for Scientific, Technical and Cultural Affairs, promoting Scientific and Technical Collaboration with Central and Eastern Europe. S. Van Huffel is a full professor with the Katholieke Universiteit Leuven. This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction (IUAP Phase V-22), initiated by the Belgian State, Prime Ministers Office-Federal Office for Scientific, Technical and Cultural Affairs, of the Concerted Research Action (GOA) projects of the Flemish Government MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology), of the IDO/99/03 project (K.U. Leuven) Predictive computer models for medical classification problems using patient data and expert knowledge, of the FWO projects G.0200.00, G.0078.01 and G.0270.02. The scientific responsibility is assumed by its authors. The authors would like to thank Maria Luisa Rastello and Amedeo Premoli for bringing the EW-TLS problem to their attention. The authors are grateful to two anonymous referees for the valuable comments.