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

TL;DR

This paper thoroughly revises and proves the consistency of the total least squares estimator in errors-in-variables linear regression models, clarifying its theoretical foundation and uniqueness.

Contribution

It provides complete proofs of TLS estimator consistency, explores its relation to eigenvalue problems, and generalizes the norm used in the estimator.

Findings

Revised consistency results for TLS estimator.

Proved the uniqueness of the TLS estimate.

Generalized the norm used in the estimator to spectral and other unitarily invariant norms.

Abstract

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. A theoretical foundation for construction of the TLS estimator and its relation to the generalized eigenvalue problem is explained. Particularly, the uniqueness of the estimate is proved. The Frobenius norm in the definition of the estimator can be substituted by the spectral norm, or by any other unitarily invariant norm; then the consistency results are still valid.