A New Perspective in Functional EIV Linear Models: Part II

• Published in: Communications in statistics. Theory and methods Vol. 50; no. 4; pp. 856 - 873
• Main Authors: Al-Sharadqah, Ali, Woolsey, Nicholas
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 16.02.2021; Taylor & Francis Ltd
• Subjects: Algorithms; Bias; bias reduction; EIV models; Least squares method; Levenberg-Marquardt algorithm; Partial differential equations; Perturbation theory; planar fitting; Weighting functions
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1080/03610926.2019.1642493

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.