Propagation of asymmetric uncertainties I have two (fully independent) measurements of the same quantity X.
Each of them reports a measurement $X_{-\sigma_L}^{+\sigma_R}$ where $\sigma_L$ and $\sigma_R$ are the left and right uncertainties (asymmetric error bars). In other words, if we call the measurements $A$ and $B$, and the subscript $A$ and $B$ stands for the measurements, we have
$A_{-\sigma_{L,A}}^{+\sigma_{R,A}}$
$B_{-\sigma_{L,B}}^{+\sigma_{R,B}}$
Now I need to calculate the different of these measurements, $\Delta = A-B$. What will be $\sigma_{L,\Delta}$, $\sigma_{R,\Delta}$? In other words, how do I propagate independent asymmetric error bars?
 A: The sum of $\chi^2$-distributed independent values is another $\chi^2$-distributed value; see https://math.stackexchange.com/questions/1096298/question-about-sum-of-chi-squared-distribution
How the values and uncertainties propagate depend on how you did the values in the first place; are they mean, mode, or median values?
The asymmetric uncertainties also depend on how they were set up. With asymmetric distributions, the confidence intervals can be set up in three different ways: (i) half of the area either side of the quoted value (eg 34% left and right of median), resulting in an asymmetric interval; (ii) a symmetric interval (eg about the median) such that the total area between the upper and lower limits is 68% - results in a symmetric interval; and (iii) upper and lower limits such that the area between them is 68% and the distance between the upper and lower is as small as possible, resulting in an asymmetric interval.
Once you now that, you can apply that to the final distribution to get the correct bounds for your estimate.
