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



Inequality-constrained linear-regression, what is the covariance of the estimator?

> 
> Well, the vector $x$ is random right? 
> 
> 
> 


It's a parameter, so therefore fixed (yet unknown): the estimator $\hat x$ is a random variable. I would agree with Brian that a covariance matrix will not be all that useful the constraints will mean that the estimator will tend to concentrate around the edges, where a lot of the asymptotic machinery breaks down.


Personally, I reckon a Bayesian approach would be better, as the inequality constraints can be easily built into the prior. 


