Int. J. Metrol. Qual. Eng.
Volume 3, Number 2, 2012
|Page(s)||117 - 123|
|Published online||14 November 2012|
Least-squares fitting with errors in the response and predictor
1 Statistical Sciences, Los Alamos
National Laboratory, USA
2 Safeguards Science and Technology, Los Alamos National Laboratory, USA
3 Department of Physics, Alma College, USA
Accepted: 22 March 2012
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 for β0 and for β1 and for an approximation to covariance (, ). 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(, ) 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.
Key words: Least square / regression / Bayesian estimation / errors
© EDP Sciences 2012
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