# Generalized least squares error estimation

First of all, I have to admit that I am not statistician so some of my nomenclature could not be very rigorous and maybe a bit confusing; pleas ask me to clarify if necessary.

The Problem

Let's say I want to solve a linear system $Ax = b$ where my vector $b$ has some uncorrelated and non-constant Gaussian noise (i.e. noise variance depends on the entry of $b$). As far as I know, the optimal way of solving this system in terms of minimum least-squares error is to use weighted least squares. Therefore, $$x_w = (A^TW^{-1}A)^{-1}A^TW^{-1}b$$ gives the optimal solution where $W$ is a diagonal matrix with the variance of each entry of $b$.

Let's also say that I can perform a linear transform $B$ to the system which will correlate my noise, $$BAx = Bb,$$ therefore now I need to solve the system using generalized least squares, $$x_g = ((BA)^TG^{-1}BA)^{-1}(BA)^TG^{-1}b,$$ where $G = BGB^T$ is the covariance matrix.

My questions

In practice, I can solve the problem in these two ways, but I do not know which solution will have lower residual, so my questions are:

• Is there any transform $B$ that will make $x_g$ have less 'error' than $x_w$?
• Are there error estimations for weighted and generalized least squares solutions? Do they depend on the patterns of $W$ and $G$? For example, a generalized least squared solution with a full $G$ matrix will have higher or lower error than a weighted least square solution where the diagonal matrix $W$ has large values (variance) at the diagonal?

What I want to do is to see if linear transforming my problem (with the colateral effect of correlating the noise) could in some cases give me better estimators of the solution.

PD: I do not ask for a long solution but just some references and guidance in order to solve this concern that I have.