  We consider the problem of detecting a `bump' in the intensity of a Poisson
process or in a density. We analyze two types of likelihood ratio based
statistics which allow for exact finite sample inference and asymptotically
optimal detection: The maximum of the penalized square root of log likelihood
ratios (`penalized scan') evaluated over a certain sparse set of intervals, and
a certain average of log likelihood ratios (`condensed average likelihood
ratio'). We show that penalizing the {\sl square root} of the log likelihood
ratio - rather than the log likelihood ratio itself - leads to a simple penalty
term that yields optimal power. The thus derived penalty may prove useful for
other problems that involve a Brownian bridge in the limit. The second key tool
is an approximating set of intervals that is rich enough to allow for optimal
detection but which is also sparse enough to allow justifying the validity of
the penalization scheme simply via the union bound. This results in a
considerable simplification in the theoretical treatment compared to the usual
approach for this type of penalization technique, which requires establishing
an exponential inequality for the variation of the test statistic. Another
advantage of using the sparse approximating set is that it allows fast
computation in nearly linear time.
  We present a simulation study that illustrates the superior performance of
the penalized scan and of the condensed average likelihood ratio compared to
the standard scan statistic.
