I'm trying to solve the following least squares problem:


$\underset{x}{\text{min}} ||Ax - \tilde{b}||\_2$


where $Ax = b$ and $\tilde{b} = b + w$


### Question:


How do I determine which probability distribution fits $w$ best?


Also, $A \in \mathbb{R}^{n\times n}$ is a large, and sparse Toeplitz matrix, $\tilde{b} \in \mathbb{R}^{n \times 1}$


$w$ is not your typical measurement noise and is probably not Gaussian. Also, I have access to $b$ (training data) and therefore, $w$.


Also, in a related question, I'm using LSQR to determine $x$ --- since I have my training data, is there any way to do cross-validation to determine how to best solve for $x$?


For example, I tried Tikhonov regularization to find the best $\lambda$ but I'm looking for a better method.



Determine noise distribution
There's no way to answer in general what distribution data has. It has whatever distribution it has, but what you really want to know is whether a distribution which someone has identified adequately fits your data. So you have to propose a specific distribution first, then test for goodness of fit. Searching on "goodness of fit" will give you lots of resources, but a good general-purposed goodness of fit test is Kolmogorov-Smirnov, often abbreviated K-S.


You could look at a histogram of your data to start your search. Is it symmetric or skewed? Long tails or short tails? Unimodal or bimodal? If you post an image, someone may be able to suggest some distributions to test.


Your question may be more appropriate for the [Statistical Analysis](https://stats.stackexchange.com/) Stack Exchange site.


