# Wald test for Poisson distributions

$$N_A$$ and $$N_B$$ are variables of the counts of the number of events 'A' and events 'B' respectively. Those variables follow Poisson distributions with parameters $$\lambda_A$$ and $$\lambda_B$$.

In nature I made one observation of each variable; I observed $$n_A$$ events 'A' and $$n_B$$ events 'B'. From those, I am asking: Are $$\lambda_A$$ and $$\lambda_B$$ equal or different?

How can I make a hypothesis testing on the null that the rates $$\lambda_A$$ and $$\lambda_B$$ are the same?

The answer suggest to use a Wald test with this formula:

$$Z=\frac{\widehat{\lambda}_1-\widehat{\lambda}_2}{\sqrt{\frac{{\widehat{\lambda}_1}}{n_1}+\frac{{\widehat{\lambda}_2}}{n_2}}}$$

Based on the comments here, I suppose $$n_1, n_2$$ are what the questioner calls $$n_A, n_B$$. But I'm not sure about $$\widehat{\lambda}_1$$ – I would assume that it is the estimate for $$\lambda_1$$, but isn't that just $$n_1$$ (as the highest probability that one sees $$n$$ occurrences is when $$\lambda = n$$)?

• Your post is highly confusing. You start by defining the $\lambda_i$ as random variables, but then you talk about the $\hat{\lambda}_i$ (i.e. as if the $\lambda$s were parameters -- which is consistent with the conventional usage). You then talk about the $n_i$ (which is defined in the question at the link you gave as observed values -- realizations of $N_1$ and $N_2$) as if they were random variables, which they aren't. Please distinguish your random variables from your parameters & your observations, and check the question at the page you link to for a definition of the notation it uses. – Glen_b Mar 11 '19 at 22:48
• I agree with @Glen_b that your question is confusing. Maybe a look at this page (or its links) will help you get oriented, and maybe provide a useful test. If you get oriented without finding a solution, perhaps edit your question so someone can help. – BruceET Mar 13 '19 at 6:33
• Thanks Glenn and Bruce – I have tried to update my question, does this help? Basically: it seems like this equation requires four variables but I only know two. – Xodarap Mar 13 '19 at 20:53
• To apply the answer you referenced, you need to take $n_1=n_2=1$ because you have exactly one observation from each distribution. – whuber Mar 13 '19 at 21:40