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In this paper, we prove strong consistency of an estimator by the truncatedsingular value decomposition for a multivariate errors-in-variables linearregression model with collinearity. This result is an extension of Gleser'sproof of the strong consistency of total least squares solutions to the casewith modern rank constraints. While the usual discussion of consistency in theabsence of solution uniqueness deals with the minimal norm solution, thecontribution of this study is to develop a theory that shows the strongconsistency of a set of solutions. The proof is based on properties oforthogonal projections, specifically properties of the Rayleigh-Ritz procedurefor computing eigenvalues. This makes it suitable for targeting problems wheresome row vectors of the matrices do not contain noise. Therefore, this papergives a proof for the regression model with the above condition on the rowvectors, resulting in a natural generalization of the strong consistency forthe standard TLS estimator.

Strong consistency of an estimator by the truncated singular value decomposition for an errors-in-variables regression model with collinearity