Scan statistic ($T={\max }_{S\in \mathcal{S}}{T}_S$)

A scan statistic searches a family of candidate locations or subsets and reports the largest local statistic. Under a null hypothesis, a union bound converts a tail estimate for each candidate into a bound for the maximum; no independence between local statistics is required. Structured alternatives often permit a chi-squared testing lower bound using the overlap geometry of two candidates.

For independent uniformly placed cyclic intervals of length $k<d$, an overlap is possible at at most $2k-1$ starting-point offsets, or at all $d$ offsets if that number exceeds $d$. Thus the overlap probability is at most $\min \{1,(2k-1)/d\}$. If $R=|S\cap T|$, then $0\le R\le k$ and $\mathbb{E}(e^{aR/k}-1)\le (2k/d)(e^a-1)$ for $a\ge 0$. The overlap law is not the hypergeometric distribution for unrestricted random subsets.

A quadratic scan statistic maximizes a directional empirical second moment over candidate unit vectors. For a Gaussian random vector sample with a rank-one covariance spike in one of those directions, the matching statistic has its scale multiplied by the spike's eigenvalue. A chi-squared concentration inequality controls each direction.