Application of least squares variance component estimation to errors-in-variables models

Abstract

In an earlier work, a simple and flexible formulation for the weighted total least squares (WTLS) problem was presented. The formulation allows one to directly apply the existing body of knowledge of the least squares theory to the errors-in-variables (EIV) models of which the complete description of the covariance matrices of the observation vector and of the design matrix can be employed. This contribution presents one of the well-known theories—least squares variance component estimation (LS-VCE)—to the total least squares problem. LS-VCE is adopted to cope with the estimation of different variance components in an EIV model having a general covariance matrix obtained from the (fully populated) covariance matrices of the functionally independent variables and a proper application of the error propagation law. Two empirical examples using real and simulated data are presented to illustrate the theory. The first example is a linear regression model and the second example is a 2-D affine transformation. For each application, two variance components—one for the observation vector and one for the coefficient matrix—are simultaneously estimated. Because the formulation is based on the standard least squares theory, the covariance matrix of the estimates in general and the precision of the estimates in particular can also be presented.