If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$

Now, what if I add the linear inequality constraints $Bx > c$? There are algorithms that find the answer for a given $e$, but what is the covariance matrix?

It seems like a non-trivial problem: http://www.gurulib.com/_user_manual_file/pic_1247578519497.pdf

However, the author seems to give up: "A much more interesting problem is to analyze a properly truncated variance-covariance matrix of $b*$. However, it is beyond the scope of this paper."

Of course, I can do a Monte-Carlo simulation, but a closed-form solution would be better. Any hint or reference?

Thanks in advance, Tony