Improving the weighted least squares estimation of parameters in errors-in-variables models

Although the weighted least squares (LS) method is straightforward to deal with errors-in-variables (EIV) models, it results in the biased estimates of parameters and the variance of unit weight. The total least squares (TLS) method is statistically rigorous and optimal but is computationally much more demanding. This paper aims at constructing an improved weighted LS estimate of parameters from the perspective of multiplicative error models, which is expected to mainly maintain the advantages of computational simplicity of the weighted LS method and the optimality of the weighted TLS method to some extent but almost remove the bias of the weighted LS estimate and free the computational burden of the TLS method. The statistical aspects of the weighted LS estimate developed from the perspective of multiplicative error models have been analyzed and an almost unbiased estimate of the variance of unit weight proposed. Finally, an N-calibrated weighted LS estimate has been constructed from the perspective of multiplicative error models.