Performance of linear least squares regression subject to inequality (bounded interval) constraints on parameters Consider the following model:
$${\bf y} = {\bf X}{\bf b} + {\bf e}$$ 
where ${\bf y}, {\bf n}\in {\cal R}^m$, ${\bf b}\in{\cal R}^n$, and ${\bf X}\in{\cal R}^{m \times n}$ where $m>n = {\rm rank}({\bf X})$. ${\bf e}$ is an error vector (with independent, identically distributed components of zero mean and variance $\sigma^2$).
Upon observation of ${\bf y}$, an estimate of ${\bf b} =[b_0, \cdots, b_{n-1}]^T$ (where $[\cdot]^T$ denotes transpose) is sought subject to bounding constraints on each parameter:
$$b_i \in [l_i,u_i]\quad i\in\{0,1,\cdots, n-1\}$$
There are numerical packages to solve such problems.
In the absence of these constraints, it is well-known that the least squares estimate of $\bf b$ is unbiased with covariance $\sigma^2 ({\bf X}^T {\bf X})^{-1}$. 
Do simple results exist for the bias and covariance of the estimation error exist under the constrained parameter scenario?  
In addition, assuming, say, normality of the error vector, are there simple expressions for the Cramer Rao Bound (CRB) of $\bf b$ under the parameter constraints?
I think I may have a solution at least for the CRB.  As per http://davegiles.blogspot.com/2015/05/maximum-likelihood-estimation.html, I could non-linearly re-parameterize the unknowns such that the constraints are automatically satisfied:   
$$b_i = f(\theta_i) = l_i + (u_i-l_i)e^{\theta_i}/(1+e^{\theta_i}).$$
The CRB of $\{\theta_i\}$ is easily calculated for normally distributed errors.  Determination of the CRB of the transformation to a "new" set of variables $\{b_i\}$ is also easy.  Any thoughts?  Does this seem correct?
 A: Your idea could work, but then the problem becomes a more generic likelihood maximization problem: it is no longer an ordinary least-squares regression and the maximization might need to be performed numerically, and edge conditions could be tricky to handle.
A better method is use quadratic programming to solve for the minimum squared residuals given your constraints. This proceeds in a manner analogous to the method of Lagrange multipliers, where an expression is extremized given an equality condition. The basic idea is to take the expression for the squared residuals and the inequalities and form their Wolf Dual. Then this dual object can be much more readily extremized. The original paper by Wolfe solves your problem explicitly and is highly regarded.
Now, regarding the covariance of the estimate: This is much trickier. It's quite likely that if the least-squares estimate of the parameter $\beta$ is outside the constraint range, the quadratic optimization value will be on the boundary. The covariance is likely to be ill-defined. There are two alternate methods you could consider: ridge regression and the lasso. There are popular packages for performing these types of regression. One interpretation of them is that they place an a priori probability density on the parameters. These probabilities are, as it happens, normal distributions and absolute-exponential distributions with means of 0, respectively, but you could redefine and adapt your problem so that either of these is suitable. Since the a priori pdfs are smooth, so are the posterior pdfs, and the covariances are well-defined.
