The H-test probability distribution revisited: Improved sensitivity

Abstract

Aims: To provide a significantly improved probability distribution for the H-test for periodicity in X-ray and $\gamma$-ray arrival times, which is already extensively used by the $\gamma$-ray pulsar community. Also, to obtain an analytical probability distribution for stacked test statistics in the case of a search for pulsed emission from an ensemble of pulsars where the significance per pulsar is relatively low, making individual detections insignificant on their own. This information is timely given the recent rapid discovery of new pulsars with the Fermi-LAT t $\gamma$-ray telescope. Methods: Approximately $10^{14}$ realisations of the H-statistic ($H$) for random (white) noise is calculated from a random number generator for which the repitition cycle is $\gg 10^{14}$. From these numbers the probability distribution $P(>H)$ is calculated. Results: The distribution of $H$ is is found to be exponential with parameter $\lambda=0.4$ so that the cumulative probability distribution $P(>H)=\exp{(-\lambda H)}$. If we stack independent values for $H$, the sum of $K$ such values would follow the Erlang-K distribution with parameter $\lambda$ for which the cumulative probability distribution is also a simple analytical expression. Conclusion: Searches for weak pulsars with unknown pulse profile shapes in the Fermi-LAT, Agile or other X-ray data bases should benefit from the {\it H-test} since it is known to be powerful against a broad range of pulse profiles, which introduces only a single statistical trial if only the {\it H-test} is used. The new probability distribution presented here favours the detection of weaker pulsars in terms of an improved sensitivity relative to the previously known distribution.