Holm Step-Down Familywise Error Control

By [F1] and [F2], each $p_i$ is measurable into $[0,1]$ with its Borel trace. For indices $a,b$, the difference $p_a-p_b$ is measurable by [F4], so $\{p_a\le p_b\}$ and $\{p_a<p_b\}$ are measurable preimages by [F5] of the Borel half-lines $(-\mathrm{\infty },0]$ and $(-\mathrm{\infty },0)$. For each permutation $\pi$, its ordering cell $C_{\pi }$ is the finite intersection over positions $a<b$ of $\{p_{\pi (a)}\le p_{\pi (b)}\}$ if $\pi (a)<\pi (b)$ and $\{p_{\pi (a)}<p_{\pi (b)}\}$ otherwise. These cells are measurable by [F6], and every $x$ belongs to exactly one because lexicographic ordering with index tie-breaking gives a unique permutation by [F3]. For rank $j$, let $Q_{\pi ,j}=\{p_{\pi (j)}\le \alpha \mathrm{/}(m-j+1)\}$; this pass event and its failure event $Q_{\pi ,j}^c$ are measurable preimages of Borel half-lines by [F1], [F2], and [F5]. For $k\in \{0,\dots ,m\}$, intersect $C_{\pi }$ with the pass events for $j\le k$ and, when $k<m$, the failure event at $k+1$. Empty intersections are omitted when $k=0$ or $k=m$. The resulting measurable cell is exactly the event that the order is $\pi$ and the returned prefix length is $k$, by [F3]; these finitely many cells partition $\mathcal{X}$.

The event $D_i(\alpha )$ is the union of the cells from step 1.1 over all permutations and prefix lengths $k$ with $i\in \{\pi (1),\dots ,\pi (k)\}$. This is a finite union because there are $m!$ permutations and $m+1$ prefix lengths. By [F6], $D_i(\alpha )\in \mathcal{A}$ for every $i$.

Fix $\theta$ and put $I_0=I_0(\theta )$ and $m_0=\mathrm{\mid }{I}_0\mathrm{\mid }$. By [F4], the Holm rejection events are measurable, so the FWER from [F2] is defined. If $m_0=0$, no true null can be rejected. Otherwise, let $j_0$ be the least rank occupied by a true null in the ordering from [F3]; it exists because $I_0$ is finite and nonempty. Since [F3] rejects only a prefix, any rejection of a true null forces rank $j_0$ to be rejected. Thus $p_{\pi (j_0)}(X)\le \alpha \mathrm{/}(m-j_0+1)$. All $m_0$ true nulls occur at ranks $j_0,\dots ,m$, so $m-j_0+1\ge m_0$ and $\alpha \mathrm{/}(m-j_0+1)\le \alpha \mathrm{/}{m}_0$. Hence any false rejection implies ${\bigcup }_{i\in I_0}\{p_i(X)\le \alpha \mathrm{/}{m}_0\}$.

If $m_0\ge 1$, fix $\theta$ and consider the fixed subfamily indexed by $I_0(\theta )$; it has $m_0$ members, all true under this $\theta$, and it is not selected from the observed p-values. By [F1] its p-values are valid. Applying the Bonferroni theorem [F5] to this subfamily at level $\alpha$ bounds the probability of its threshold-union event ${\bigcup }_{i\in I_0}\{p_i(X)\le \alpha \mathrm{/}{m}_0\}$ by $\alpha$. This contains the Holm false-rejection event from step 1.1, so ${\mathrm{FWER}}_{\theta }\le \alpha$. If $m_0=0$, step 1.1 gives FWER zero. Since $\theta$ was arbitrary, the control is strong.