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We consider the problem of detecting a `bump' in the intensity of a Poissonprocess or in a density. We analyze two types of likelihood ratio basedstatistics which allow for exact finite sample inference and asymptoticallyoptimal detection: The maximum of the penalized square root of log likelihoodratios (`penalized scan') evaluated over a certain sparse set of intervals, anda certain average of log likelihood ratios (`condensed average likelihoodratio'). We show that penalizing the {\sl square root} of the log likelihoodratio - rather than the log likelihood ratio itself - leads to a simple penaltyterm that yields optimal power. The thus derived penalty may prove useful forother problems that involve a Brownian bridge in the limit. The second key toolis an approximating set of intervals that is rich enough to allow for optimaldetection but which is also sparse enough to allow justifying the validity ofthe penalization scheme simply via the union bound. This results in aconsiderable simplification in the theoretical treatment compared to the usualapproach for this type of penalization technique, which requires establishingan exponential inequality for the variation of the test statistic. Anotheradvantage of using the sparse approximating set is that it allows fastcomputation in nearly linear time.  We present a simulation study that illustrates the superior performance ofthe penalized scan and of the condensed average likelihood ratio compared tothe standard scan statistic.

Optimal detection of a jump in the intensity of a Poisson process or in a density with likelihood ratio statistics