# Test if one Poisson process's rate is smaller than a given value

Suppose $N(t)$ is a Poisson process with rates $\lambda$. Suppose I've been observing it for $t \in [0, T]$ and recorded events. How can I test the null hypothesis $\lambda < \lambda_0$, where $\lambda_0$ is a given value?

An obvious solution is to derive MLE of rates $\hat\lambda$ and compare with $\lambda_0$. But I don't know how to calculate the statistical significance then. Any idea?

[Updated the question on Jan 4.]

• I don't know what would be the "best," because I'm not sure what your objectives, constraints, or situation might be, but it occurs to me that you could easily exploit the fact that under the null hypothesis $\lambda_1=\lambda_2$, $N_1(t)$ has a Binomial$(N,1/2)$ distribution conditional on $N=N_1(t)+N_2(t)$. Thus a one-tailed Test of Proportions would work.
– whuber
Jan 4, 2017 at 20:37
• @whuber Good point. By best I mean "principled", or probably just "well known". But the null hypothesis isn't exactly $\lambda_1 = \lambda_2$.. Jan 4, 2017 at 20:45
• That makes no difference: I'm sure you're aware the Test of Proportions handles both one-sided and two-sided tests and in either case the distribution under the null is the one determined by the situation where $\lambda_1=\lambda_2$. You may specify more general hypotheses, such as $\lambda_1 \ge 2\lambda_2$ for instance, and all that would change is the Binomial probability you use in the test.
– whuber
Jan 4, 2017 at 21:05
• @whuber So, under "my" null, events generated by $N_1$ should have a smaller proportion. Is that the idea? I'd still appreciate if you could give more details. It might seem trivial to you but not to me. Jan 4, 2017 at 21:34
• stats.stackexchange.com/search?q=test+of+proportions
– whuber
Jan 4, 2017 at 21:35