How do I compare two event rates?

Say I have two event streams, modeled as Poisson processes. These streams omit events at rates $\lambda_1$ and $\lambda_2$ respectively, so the count of observed events in $t$ time is $\lambda_1t$ and $\lambda_2t$. For some desired confidence, how long do I need to observe in order to be confident that either $\lambda_1 > \lambda_2$ or vice versa?

Intuition suggests that the answer will be some distribution that's a function of $\lambda_1$, $\lambda_2$, $t$, and my desired confidence. I did some searching and just found tests for inequality of $\lambda_1$ and $\lambda_2$, not a specific parameterized distribution.

• You need to decide on a parametrization appropriate for what you're modelling. Often the parameter of interest would be a function of either $\lambda_1 - \lambda_2$ or $\frac{\lambda_1}{\lambda_2}$, with $\lambda_1+\lambda_2$ being a nuisance parameter. I've added the power-analysis tag. Commented Jun 1, 2018 at 11:13
• Thanks. I'm not sure which parameterization is appropriate. I'm just trying to detect whether $\lambda_1$ and $\lambda_2$ are different. Ideally, I could detect whether any of $\lambda_1, \lambda_2, ..., \lambda_n$ are greater than the others.
– tom
Commented Jun 1, 2018 at 14:31
• You'll need to wait longer to detect a difference of one event per hour than to detect a difference of a hundred events per hour. Typically you'd set the Type I error of your test - the probability of rejecting the null hypothesis of no difference when it is in fact true - to 1%, say, & the power - the probability of rejecting the null hypothesis when a given alternative is in fact true - to 90%, say, for an effect size of your choosing; & then solve for the required observation time. So you do need to pick a particular parametrization first. Commented Jun 1, 2018 at 15:10
• You'll also - not that this need be given in your question - have to make a guess, typically a pessimistic one, at the overall event rate (or one of the individual rates). Clearly if you expect to wait a day or so to observe any events at all across both streams, the study will take longer than if you expect many over the course of an hour. Commented Jun 1, 2018 at 15:30
• @Scortchi That was my thought as well...pick a Type 1 error and Type 2 error tolerance, and solve for t. I was hoping someone would say "Oh, that's just a XYZ distribution parameterized as follows", but it looks like I may have to do the math myself.
– tom
Commented Jun 1, 2018 at 20:30