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Least-squares fitting with errors in the response andpredictor

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:[email protected]
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Abstract

Least squares regression is commonly used in metrology for calibration and estimation. Inregression relating a response y to a predictor x, thepredictor x is often measured with error that is ignored in analysis.Practitioners wondering how to proceed when x has non-negligible errorface 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}} $}β̂1, EIP for β1 and for anapproximation to covariance (\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}β̂0, EIP, \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}β̂1, EIP). It is assumed that there are measured datax = xtrue + ex,andy = ytrue + eywith errors ex in x andey in y and thevariances of the errors ex andey are allowed to depend onxtrue and ytrue, respectively.This paper also investigates the accuracy of the estimated cov(\hbox{$\Hat{{\beta }}_{0, {\rm EIP}} $}β̂0, EIP, \hbox{$\Hat{{\beta }}_{1, {\rm EIP}} $}β̂1, EIP) and provides a numerical Bayesian alternative usingMarkov Chain Monte Carlo, which is recommended particularly for small sample sizes wherethe 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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