# Propagating asymmetric and large errors in a ratio

I'm currently trying to propagate the errors of two values:

$A=143^{+28.6}_{-26.6}$ , $B=19^{+10.9}_{-8.7}$ $\rightarrow$ $B/A = 0.133^{+?}_{-?}$

Both values A and B (#counts) are derived from a log-likelihood model fit on poisson data. The asymmetrical errors are due to the low number of events, which means that the Poisson-$\chi^2$ distribution (and thus the errors) is slightly asymmetric. I also know the numerical chi2 (likelihood) distribution for A and B.

Now, there are multiple major problems that I'm facing in the error propagation:

1) The errors are asymmetric, which means that the "normal" calculations for propagation won't work.

2) The errors are large (50%), which also breaks standard error propagation. Could be circumvented by using Fieller's theorem?

3) Even if you consider the errors as symmetric and small, calculating the ratio of two gaussian distributions can still result in an asymmetric ratio distribution.

The major struggle that I have right now is to incorporate all of those three issues into one method. If this is very time demanding, I could also assume the errors to be symmetric (because the difference isn't sooo big). However, even then, the problem of 2) and 3) still exist.

One solution that I've thought of is a Toy-MC:

You calculate $B/A$ based on random B's and A's that are generated based on the probability of B&A (probability calculated with the Chi2=likelihood distribution). However, I'm totally not sure if this is correct (and also how you would get the errors from the B/A distribution then).

• I just realized that Fieller's theorem would also not help, because it is meant to be used on the mean of some sample data (like in biology or medicine) and not on values that are fit based on some data. I.e. I have a certain number of low statistics "counts" in data and then I fit the data with some model in order to derive the averaged #counts/time-unit. – 0vbb Jan 13 '17 at 3:12
• Cross-posted to Physics: physics.stackexchange.com/q/304762/44126 – rob Jan 13 '17 at 4:19