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Estimate the signal by one-dimensional total variation denoising, which penalizes adjacent differences rather than the constant level:This is a fused Lasso objective. Its L1 norm penalty makes many adjacent differences exactly zero, producing a piecewise-constant fit. For change-point detection, estimate the locations by $\hat{S}=\{i:{\hat{\mu }}_i\ne {\hat{\mu }}_{i+1}\}$. The tuning parameter controls the strength of the penalty; it can be chosen, for example, using a validation criterion appropriate to the noise model.

For a general family of elementary nulls $H_1,\dots ,H_m$, the closed testing procedure forms every nonempty intersection $H_B={\bigcap }_{j\in B}{H}_j$ and gives it a local test of level $\alpha$. Reject $H_j$ precisely when all intersections with $j\in B$ are locally rejected. This gives strong control of the familywise error rate: let $T_0$ be the indices of the true elementary nulls. If $T_0$ is empty there is no false rejection. Otherwise, rejecting any $H_j$ with $j\in T_0$ requires rejection of the true intersection $H_{T_0}$. Its local test has rejection probability at most $\alpha$, henceThis proof of closed-testing control of the familywise error rate needs no independence of the tests.

For the interval procedure, a true tested interval is contained in one maximal constant block $B\in \mathcal{T}$. If $I\subseteq B$, the intervals over which $p_B$ takes its maximum are a subset of those used for $p_I$, so $p_I\ge p_B$. Therefore rejection of a true $I$ forces rejection of its containing block. Conversely, a rejected block is itself a rejected true null.

There is a small indexing issue in the PDF: its testing family excludes singleton intervals, while $\mathcal{T}$ may contain singleton blocks. Put ${\mathcal{T}}_{\ast }=\{B\in \mathcal{T}:|B|\ge 2\}$. Every true tested interval lies in one of these blocks, so the exact statement, using only hypotheses defined in the paper, isSingleton blocks contain no eligible tested interval and can be ignored. Equivalently, one may define their nulls as never rejected.

For each $B\in {\mathcal{T}}_{\ast }$, the term $J=B$ occurs in the maximum defining $p_B$, giving $p_B\ge n{q}_B/|B|$. A valid p-value is a super-uniform random variable under its null, so for $0\le \alpha \le 1$,The union bound and disjointness of the constant blocks now giveThis establishes weighted interval testing for a piecewise-constant mean even when the interval p-values are dependent. Values $p_I>1$ do not affect rejection at levels at most one.

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.

For a two-by-two contingency table, Fisher's exact test conditions on both row totals and both column totals. Under the null of no association, the upper-left count $K$ then has the hypergeometric distributionwhere $r_1$ and $c_1$ are the first row and column totals. A one-sided p-value sums the appropriate hypergeometric tail; a common two-sided p-value sums the probabilities of all feasible tables no more likely under the null than the observed table. Because the conditional distribution is discrete, the p-value is super-uniform random variable rather than generally exactly uniform.