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I have an estimator $\hat{R}$ that I am computing for $N$ groups. For each group I am performing a hypothesis test with significance ($\alpha$) : $H_0$: $\mathbb{E}[\hat{R}]=r$, $H_1$: $\mathbb{E}[\hat{R}]\neq r$ where $r$ is just an expected number that I provide. Because I am doing $N$ hypothesis test, if $H_0$ is true for all groups, I would still expect $N\alpha$ type I errors where the null hypothesis in incorrectly rejected. In order to correct for this, one can use something like a Bonferroni correction and use a significance of $\alpha^*=\alpha/N$ instead for all test.

QUESTION: Is using the Bonferroni correction equivalent in some form to performing a second-step hypothesis testing where after performing the $N$ hypothesis tests mentioned above, one performs another test where $H_0$ is now: # of violated hypothesis = $\alpha N$ ? If so, is there a name for this approach ?

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The idea behind Bonferroni is simply how you already explained: The probability of wrongly rejecting $H_0$ in one of your tests increases linearity with the number $N$ of your tests, so we just make it $N$ times harder to wrongly reject the null.

And you are right, if you want to, you can think of this as wrapping your $N$ hypothesis tests into another test (a binomial hypothesis test).

But note that the method of Bonferroni correction is rather old and better methods have been developed since. I'd recommend to use newer methods like e.g. the much much cooler Benjamini-Hochberg procedure.

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