A New Perspective in Functional EIV Linear Models: Part II

Abstract

In Error-in-Variables (EIV) models, all variables are subject to error and their estimators are heavily biased especially in the case of small sample sizes. This paper extends the results of Al-Sharadqah and proposes a new estimator for the problem of fitting 3D planes to data. Instead of minimizing an objective function obtained by the likelihood principles or others, a family of objective functions depending on an unknown smooth weight is considered. Then, the optimal weight function is derived so that the minimizer of the corresponding objective function has a zero second-order bias. To derive such a weight, a general form of the second-order bias is derived by applying our perturbation theory. This leads to a system of two first-order linear partial differential equations that admits a unique solution. The explicit formula of the weight is derived yielding the objective function. This turns to be a standard nonlinear least squares problem. Accordingly, the Levenberg-Marquardt (LM) algorithm is implemented. Finally, the effectiveness and superiority of our method are assessed by a series of numerical experiments.

Keywords:

• EIV models
• bias reduction
• planar fitting
• Levenberg-Marquardt algorithm

Acknowledgment

The authors would like to thank the reviewers and the editor for providing many suggestions that resulted in improving this paper.

Notes

1 Be aware that the normalized KCR here is for $\widehat{\Theta }={(\widehat{\alpha },\widehat{\beta },\widehat{\gamma })}^{\top }$ not $\widehat{\mathit{\theta }}={(\widehat{\alpha },\widehat{\beta })}^{\top }.$