Consistent estimation with the use of orthogonal projections for a linear regression model with errors in the variables

Abstract

In this paper, we construct an estimator of a particular errors-in-variables linear regression model. It is well known that the total least squares yield a consistent estimator of the ordinary errors-in-variables linear regression model. We aim to extend the above consistency in the statistical sense. More specifically, we consider the case where, in an errors-in-variables linear regression model, certain rows and columns do not contain errors for their matrix representation. Such a regression model leads to a constrained total least squares problems with row and column constraints. Although this problem can be solved numerically, it is not known whether the solution has consistency in the statistical sense in the previous study. With such a research background, we propose an estimator constructed by using orthogonal projections and their properties, so that its strong consistency is naturally proved. Moreover, our asymptotic analysis with emphasis on orthogonal projections proves the strong consistency of the total least squares solution of the problem with row and column constraints.