  We consider the problem of detecting a rectangle of activation in a grid of
sensors in d-dimensions with noisy measurements. This has applications to
massive surveillance projects and anomaly detection in large datasets in which
one detects anomalously high measurements over rectangular regions, or more
generally, blobs. Recently, the asymptotic distribution of a multiscale scan
statistic was established in (Kabluchko, 2011) under the null hypothesis, using
non-constant boundary crossing probabilities for locally-stationary Gaussian
random fields derived in (Chan and Lai, 2006). Using a similar approach, we
derive the exact asymptotic level and power of four variants of the scan
statistic: an oracle scan that knows the dimensions of the activation
rectangle; the multiscale scan statistic just mentioned; an adaptive variant;
and an epsilon-net approximation to the latter, in the spirit of (Arias-Castro,
2005). This approximate scan runs in time near-linear in the size of the grid
and achieves the same asymptotic power as the adaptive scan. We complement our
theory with some numerical experiments.
