Assume that I have a system and that I can measure both the inputs and outputs of and that I can formulate a solution I want in the form of linear regression: $A \vec{x}=\vec{b}$. Further assume that I've measured $A$ and $\vec{b}$ twice. The first time I measure everything, $A_1 \in \mathbb{R}^{450 \times 45}$ and $\vec{b}_1 \in \mathbb{R}^{450 \times 1}$. The second time, $A_2 \in \mathbb{R}^{450 \times 45}$ and $\vec{b}_2 \in \mathbb{R}^{450 \times 1}$. The entries in the $A$ and $\vec{b}$ are not the same.
I can solve both of these systems of equations to get $\vec{x}_1$, $\Sigma_{x_1}$, $\vec{x}_2$, and $\Sigma_{x_2}$. Where the $\Sigma$ are the variance-covariance matrices for the solutions, $\vec{x}$. I can think of two ways to argue that $\vec{x}_2$ is likely to be different from $\vec{x}_1$.
Compute the euclidean distance between $\vec{x}_1$ and $\vec{x}_2$ and then do a Welch t-test on the distance. This always shows extreme significance. I think because of the curse of dimensionality.
Compute how (un)likely it would be to draw $\vec{x}_2$ from the distribution defined by $\vec{x}_1$ (and its $\Sigma$). The likelihood is just the integral of all space as close to $\vec{x}_1$ as $\vec{x}_2$ or further away. This seems asymmetric, so do the same for $\vec{x}_1$ drawn from $\vec{x}_2$ (and its $\Sigma$) and average them. This seems less likely to suffer from the curse of dimensionality because I am weighting most of the space by practically zero.
Both of these methods function in that I can literally compute the results, but I've made them up ad hoc and I'd like to know if there are more established ways of computing/showing/determining/arguing that the vectors $\vec{x}_1$ and $\vec{x}_2$ are different in a statistical sense.
