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Least-squares fitting with errors in the response and predictor

Published online by Cambridge University Press:  14 November 2012

T. Burr ,
S. Croft  and
B.C. Reed
T. Burr*
Affiliation:
Statistical Sciences, Los Alamos National Laboratory, USA
S. Croft
Affiliation:
Safeguards Science and Technology, Los Alamos National Laboratory, USA
B.C. Reed
Affiliation:
Department of Physics, Alma College, USA
*
⋆ Correspondence: tburr@lanl.gov
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Abstract

Least squares regression is commonly used in metrology for calibration and estimation. In regression relating a response y to a predictor x, the predictor x is often measured with error that is ignored in analysis. Practitioners wondering how to proceed when x has non-negligible error face a daunting literature, with a wide range of notation, assumptions, and approaches. For the model ytrue = β0 + β1   xtrue, we provide simple expressions for errors in predictors (EIP) estimators \hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}β̂0, EIP for β0 and \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}β̂1, EIP for β1 and for an approximation to covariance (\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}β̂0, EIP, \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}β̂1, EIP). It is assumed that there are measured data x = xtrue + ex, and y = ytrue + ey with errors ex in x and ey in y and the variances of the errors ex and ey are allowed to depend on xtrue and ytrue, respectively. This paper also investigates the accuracy of the estimated cov(\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}β̂0, EIP, \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}β̂1, EIP) and provides a numerical Bayesian alternative using Markov Chain Monte Carlo, which is recommended particularly for small sample sizes where the approximate expression is shown to have lower accuracy than desired.

Keywords

Least squareregressionBayesian estimationerrors
Type
Research Article
Information
International Journal of Metrology and Quality Engineering , Volume 3 , Issue 2 , 2012 , pp. 117 - 123
Copyright
© EDP Sciences 2012

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