This is my first question on Math Exchange so my apologies if it does not initially fit the format.
I have a problem where I'd like to calculate the a-posteriori variance/covariance matrix of parameters produced by the $L_1$ estimator.
In comparison to the $L_2$ estimator (least squares) on the estimation of $x$ in the problem $Ax=l$ which has an analytic solution $x=(A^TPA)^{-1}A^TPl$ and an a-posteriori covariance matrix equal to $Q_{xx}=(A^TPA)^{-1}$, I'd like to find $Q_{xx}$ for the $L_1$ estimator which has no analytic solution (to my knowledge) but is rather solved using linear programming.
$P=Q^{−1}$ is the weight matrix, where $Q$ is typically a block-diagonal matrix of a-priori variance/covariances. $A$ is the Jacobian of observation equations, $l$ are the observation values. $x=(A^TPA)^{−1}A^TPl$ are the normal equations for weighted least squares. The $L_{1}$ estimator would take a different approach, whereby a linear program is formulated to minimise $\|P^{\frac{1}{2}}Ax−P^{\frac{1}{2}}l\|_1$.
Is there a way to propagate variance in an estimator where there is no analytic solution?