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I would like to model a set of data that have been measured with a certain random experimental error. Suppose $X_i,Y_i~,~i=1,...,N$ are, respectively, $\mathcal{N}(\mu_i,\sigma_x)$ and $\mathcal{N}(\nu_i,\sigma_y)$ distributed random variables and they are mutually independent (the joint distribution of all $2N$ variables factorizes). These random variables represent two sets of data coming from independent experimental measurements with some uncontrollable random errors, modeled as a small deviation from a central value to match the presentation of experimental results in the form $(x_i\pm\sigma_x, y_i\pm\sigma_y)$. A potential error function can be written as $$E(a)=\sum_{i=1}^N(Y_i-aX_i-b)^2$$

Is there any known way I can extract (approximately when the spread of the variables are small compared to their means or exactly) an error in my estimate of the most optimal regression coefficients? References to existing bibliography would be much appreciated.

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I don't have enough reputations to comment, so I express my views in the form of "answer".

What strikes me as strange is how your errors are modelled. Normally, measurement errors are modelled as $Z\ =\ Z^{*} \ +\ \eta ,\ \eta \sim N( \mu _{\eta } ,\ \sigma _{\eta })$, where $\ Z^{*}$ is the true measurement of the variable. This measurement error model can be applied on dependent and/or independent variable. If this is the measurement error model you could accept, there are numerous references available for the context of regressions:

  1. Gillard 2006
  2. Pischke 2007
  3. Wikipedia: Errors in Variables
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