Preprint #96-19



Generalization of the TLS Approach in the Errors-in-Variables Problem

by

Yasuo Amemiya


Abstract

The total least squares (TLS) method is an appropriate estimation procedure for the errors-in-variables model with a single linear relationship and with the error covariance matrix either known or known up to a multiple. This type of the errors-in-variables model can be extended in a number of ways keeping the applicability of the basic TLS approach in some form. The extensions considered here include models with a nonlinear relationship, with multiple relationships, and with an unknown error covariance matrix. Proper estimation procedures for the extended models can be considered generalizations of the basic TLS methods. For the models with a known error covariance matrix and with either a nonlinear relationship or multiple linear relationships, the estimation procedures are discussed from the TLS point of view. For the factor analysis model with multiple linear relationships and an unknown diagonal error covariance matrix, it is described how a proper estimation method can be carried out using an iterated version of the weighted TLS method with iteratively estimated weights. Also, the nonlinear factor analysis problem, an area of current statistical research interest, is discussed. An estimation procedure which can be considered a generalization of the TLS method is presented for the nonlinear factor analysis model with multiple nonlinear relationships and an unknown diagonal error covariance matrix.


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