Weighted Bonferroni correction

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.