I found a problem, which says

Let $X_1,...,X_{n_1}\sim Poisson(\lambda_1), Y_1,...,Y_{n_2}\sim Poisson(\lambda_2), i.i.d$ and independent of each other. $H_0:\lambda_1=\lambda_2, \ H_1: not \ H_0$.

Derive the Wald's test of size $\alpha$ for testing the hypothesis, when $n_1,n_2$ are large."

I've only learned about Wald's test for random samples from 1 distribution, but I don't know how to do it when comparing 2 distributions.

And I also couldn't understand what 'large' means, because even if $n_1,n_2$ both go to infinity, the speed can be different and there was no information about that.


Since to compare means of two iid distributions, you can compute the differences between respective random variables in both samples and generate new hypotheses as,

$H_0$: $\delta\ = 0\ $ and the alternate, $H_1$: $\delta\ \ne 0\ $.

And then you would get one distribution and then you can apply the wald's test on that distribution as,

computing $estimated \ \delta\\ $ using MLE and computing the standard normal RV,

$W\ = \frac{est.\ \delta -\delta_0}{est.\ se(\ est. \ \delta\ )}\ $

And finally compare it with the given $\alpha$ value for the test.

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