 Solution

The familywise error rate isThe Bonferroni correction rejects $H_i$ when $p_i\le \alpha /m$. Since a valid p-value is super-uniform under its null, the union bound givesNo independence assumption is needed.

For the closed testing procedure, form the intersection hypothesis $H_I={\bigcap }_{i\in I}{H}_i$ for every nonempty $I\subseteq \{1,\dots ,m\}$ and choose a level-$\alpha$ local test for each $H_I$. Reject $H_i$ exactly when every $H_I$ with $i\in I$ is rejected by its local test. If any true $H_i$ is rejected, then the intersection $H_{I_0}$ of all true nulls is rejected. Since its local test has significance level $\alpha$,This is the closed-testing control of the familywise error rate.

Write $W={\sum }_{j=1}^m{w}_j$. Procedure (A) is the Weighted Bonferroni correction: it rejects $H_i$ when $p_i\le \alpha w_i/W$. Therefore

Procedure (B) is the Weighted Holm step-down procedure. Let $k$ be the first rank in the ordering $q_{(1)}<\cdots <q_{(m)}$ whose hypothesis is a true null, and put $W_0={\sum }_{i\in I_0}{w}_i$. If any true null is rejected, the procedure reaches step $k$ andbecause every true null remains among ranks $k,\dots ,m$. Henceand another union bound gives FWER at most $\alpha$.

At step $k$, procedure (B) divides by the total weight still under consideration, which is no larger than $W$. Its critical values therefore increase as hypotheses are rejected. Moreover, if procedure (A) would reject a hypothesis, every earlier ordered $q$ also passes the initial threshold, so procedure (B) reaches and rejects it. Thus (B) contains every rejection of (A) and can make strictly more rejections while retaining the same strong FWER control.