The Holm-Bonferroni step-down procedure, introduced by Sture Holm in 1979, is a sequentially rejective multiple-testing method that strongly controls the family-wise error rate at level $\alpha$ while being uniformly more powerful than the single-step Bonferroni correction. Given $m$ hypotheses with p-values $p_1, \dots, p_m$, sort them in ascending order as $p_{(1)} \leq p_{(2)} \leq \dots \leq p_{(m)}$ with corresponding hypotheses $H_{(1)}, \dots, H_{(m)}$. For $i = 1, 2, \dots, m$ in turn, compare $p_{(i)} \leq \frac{\alpha}{m - i + 1}$. If the inequality holds, reject $H_{(i)}$ and proceed to step $i+1$. At the first index $i$ where the inequality fails, stop and retain $H_{(i)}, \dots, H_{(m)}$. The thresholds increase from $\alpha/m$ at step 1 to $\alpha$ at step $m$, so any hypothesis rejected by Bonferroni is also rejected by Holm, but Holm can reject additional hypotheses once earlier ones have been rejected. The procedure controls $\mathrm{FWER} \leq \alpha$ strongly under any joint distribution of the test statistics.