EIV regression with bounded errors in data: total ‘least squares’ with Chebyshev norm

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

“…For the matrix ∆ that is a solution to minimization problem (7), consider the rowspace span (C − ∆) ⊤ of the matrix C − ∆. Its dimension does not exceed n. Its orthogonal basis can be completed to the orthogonal basis in R n+d , and the complement consists of n + d − rk(C − ∆) ≥ d vectors.…”

“…(Otherwise, if the lower block of the matrix X ext is singular, then our estimation fails. Note that whether the lower block of the matrix X ext is singular might depend not only on the observations C, but also on the choice of the matrix ∆ where the minimum in (7) in attained and the d vectors that make matrix X ext . We will show that the lower block of the matrix X ext is nonsingular with high probability regardless of the choice of ∆ and X ext .)…”

“…Possible problems that may arise in the course of solving the minimization problem (7) are discussed in [18]. We should mention that our two-step definition (7) & (9) of the TLS estimator is slightly different from the conventional definition in [20, Sections 2.3.2 and 3.2] or in [10]. In these papers, the problem from which the estimator X is found is equivalent to the following:…”

“…where the optimization is performed for ∆ and X that satisfy the constraints in (10). If our estimation defined with (7) and (9) succeeds, then the minimum values in (7) and (10) coincide, and the minimum in (10) is attained for (∆, X) that is the solution to (7) & (9). Conversely, if our estimation succeeds for at least one choice of ∆ and X ext , then all the solutions to (10) can be obtained with different choices of ∆ and X ext .…”

“…However, strange things may happen if our estimation always fails. Besides (7), consider the optimization problem…”