  We investigate the performance of the scan (maximum likelihood ratio
statistic) and of the average likelihood ratio statistic in the problem of
detecting a deterministic signal with unknown spatial extent in the
prototypical univariate sampled data model with white Gaussian noise. Our
results show that the scan statistic, a popular tool for detection problems, is
optimal only for the detection of signals with the smallest spatial extent. For
signals with larger spatial extent the scan is suboptimal, and the power loss
can be considerable. In contrast, the average likelihood ratio statistic is
optimal for the detection of signals on all scales except the smallest ones,
where its performance is only slightly suboptimal. We give rigorous
mathematical statements of these results as well as heuristic explanations
which suggest that the essence of these findings applies to detection problems
quite generally, such as the detection of clusters in models involving
densities or intensities or the detection of multivariate signals. We present a
modification of the average likelihood ratio that yields optimal detection of
signals with arbitrary spatial extent and which has the additional benefit of
allowing for a fast computation of the statistic. In contrast, optimal
detection with the scan seems to require the use of scale-dependent critical
values.
