Bonferroni correction

The Bonferroni correction tests each of $m$ hypotheses at level $\alpha /m$. The union bound then controls the familywise error rate by $\alpha$ under arbitrary dependence.

For positive deterministic weights $w_i$, the weighted Bonferroni correction rejects $H_i$ whenThe union bound over the true nulls controls the familywise error rate without an independence assumption.

Order $q_i=p_i/w_i$ increasingly. At rank $k$, reject the corresponding hypothesis whenand otherwise stop. The first ordered true null reduces FWER control to a weighted union bound; the shrinking denominator makes this procedure at least as powerful as the Weighted Bonferroni correction.

The Holm–Bonferroni method orders p-values and compares the $j$th smallest with $\alpha /(m-j+1)$, stopping at the first failed comparison. It controls the familywise error rate under arbitrary dependence.

If the Holm step-down procedure makes a false rejection, it must reject the first true null in the ordered list. All $m_0$ true nulls remain at that step, so its cutoff is at most $\alpha /m_0$. The union bound on their marginally valid p-values gives familywise error rate at most $\alpha$. This proves strong control under arbitrary dependence, and super-uniformity suffices.