# Linear regression with normally distributed data and model with arbitrary covariance

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 $$vec[\Delta A, \Delta b] \sim \mathcal{N}(0,\Sigma)$$?

• 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?? – whuber Mar 26 at 11:28
• 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?). – fhchl Mar 26 at 12:04

There exists a paper by van Huffel about the generalized total least squares estimator, where the entries of $$A$$ can be correlated.