I'm collecting data for two Poisson random variables, that are completely independent of each other. My current analysis method yields 95% (or 90% or other) confidence intervals for the two $\lambda_1$, $\lambda_2$ for the two Poissons.
My question: how can I come up with a confidence interval for the ratio of the two lambdas?
You can calculate the CV for each of the two variables ($1/\sqrt{\lambda_1}$) and then add the squared CVs $CV=\sqrt{CV_1^2+CV_2^2}$.
But what you probably want to know is the confidence interval, and here the SE is not helpful as the ratio is not normally distributed. So unless the number are very large (say 1000 or so) I would rather do a parametric bootstrap.
If $\lambda_1$ belongs to $[a_1,b_1]$ with probability at least $p_1$ and $\lambda_2$ belongs to $[a_2,b_2]$ with probability at least $p_2$ for some $0<a_1<b_1$ and $0<a_2<b_2$, then, by independence, $\lambda_1/\lambda_2$ belongs to $[a_1/b_2,b_1/a_1]$ with probability at least $p_1p_2$.
For example, $p_1=p_2=90\%$ yields $p_1p_2=81\%$.
