Least-squares fitting with errors in the response and predictor

摘要

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 y_(true) = β_0 + β_1 x_(true), we provide simple expressions for errors in predictors (EIP) estimators β_(0,EIP) for β_0 and β_(1,EIP) for β_1 and for an approximation to covariance (β_(0,EIP),β_(1,EIP)). It is assumed that there are measured data x = x_(true) + e_x, and y = y_(true) + e_y with errors e_x in x and e_y in y and the variances of the errors e_x and e_y are allowed to depend on x_(true) and y_(true), respectively. This paper also investigates the accuracy of the estimated cov(β_(0,EIP),β_(1,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.