Conditions for the consistency of the total least squares estimator in an errors-in-variables linear regression model

Abstract

Abstract. A homoscedastic errors-in-variables linear regression model is considered. The total least squares estimator is studied. New conditions for the consistency and strong consistency of the total least squares estimator are proposed. These conditions are weaker than those proposed by Kukush and Van Huffel (Metrika 59 (2004), 75-97).

Consistency of the total least squares estimator in the linear errors-in-variables regression

“…The key point of the proof is the application of our own theorem on perturbation bounds for generalized eigenvectors (Theorems 6.5 and 6.6, see also [18]). The conditions were relaxed by renormalization of the data.…”

“…The errors are assumed to have the same covariance matrix for each observation and to be independent between different observations, however some variables may be observed without errors. Detailed proofs of the consistency theorems for the TLS estimator, which were first stated in [18], are presented. It is proved that that the final estimator X for explicit-notation regression coefficients (i.e., for X 0 in (1) or (2), and not the estimator X ext for X 0 ext in equation (3), which sets the relationship between the regressors and response variables implicitly) is unique, either with high probability or eventually.…”

“…The equalities (18) and (19) can be verified directly; and the symmetry properties can be reduced to the equality…”

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This paper deals with a homoskedastic errors-in-variables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems.

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“…Sufficient conditions for consistency of the estimator are presented in Gleser [5], Gallo [4], Kukush and Van Huffel [10]. In [18], the consistency results are obtained under less restrictive conditions than in [10]. In particular, there is no requirement that…”