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Consider the linear regression problem $$(A+\Delta A)x = b + \Delta b.$$

If $\Delta A = 0$ and $\Delta b$ is identically and independently distributed, then ordinary least-squares gives a good (BLUE) estimate for $x$. If $\Delta A = 0$ and $\Delta b$ has a given general covariance, then generalized least-squares gives a good estimate (BLUE). If there is error in both $A$ and $b$ such that $vec[\Delta A, \Delta b]$ are identically and independently distributed, then total least-squares give a good (strongly consistent) estimator of $x$.

What are good/efficient estimators in the general case that $vec[\Delta A, \Delta b] \sim \mathcal{N}(0,\Sigma)$?

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    $\begingroup$ If $b$ is iid then necessarily $A$ is a single column of constant values. It looks like you are consistently confusing assumptions about $b-Ax$ with assumptions about $b,$ but because of that your question is unclear. Also, under what conceivable circumstances would the entries of $A$ and $b$ be iid?? $\endgroup$ – whuber Mar 26 at 11:28
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    $\begingroup$ Yes, thank you for helping clarifying the issue! I was confusing assumptions on the errors with assumptions on the data themselves (facepalm). I have edited the question. The errors $\Delta A$ and $\Delta b$ can be iid, for example, if they where generated by the same process (which now makes sense?). $\endgroup$ – fhchl Mar 26 at 12:04
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There exists a paper by van Huffel about the generalized total least squares estimator, where the entries of $A$ can be correlated.

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