重新审视H检验概率分布：提高灵敏度
The H-test probability distribution revisited: Improved sensitivity
摘要：
目的：为X射线和$γ$射线到达时间的周期性H检验提供一个显著改进的概率分布，该概率分布已被$γ$射线脉冲星界广泛使用。此外，为了获得堆叠的测试统计量的分析概率分布的情况下，搜索脉冲发射从一个合奏的脉冲星，其中每个脉冲星的显着性是相对较低的，使个人的检测本身无关紧要。由于最近用费米-LAT t $γ$射线望远镜迅速发现了新的类星体，这一信息是及时的。研究方法：随机（白）噪声的H统计量（$H$）的近似$10^{14}$实现是由重复周期为$\gg 10^{14}$的随机数发生器计算的。根据这些数字计算概率分布$P(>H)$。结果如下：发现$H$的分布与参数$λ=0.4$呈指数关系，使得累积概率分布$P(>H)=\exp{(-λH)}$。如果我们堆叠$H$的独立值，则这些值的$K$之和将遵循具有参数$λ$的Erlang-K分布，其中累积概率分布也是简单的解析表达式。结论：对于Fermi-LAT、Agile或其他X射线数据库中具有未知脉冲轮廓形状的弱散射体，应受益于{ \it H-检验}，因为已知它对宽范围的脉冲轮廓是有效的，如果仅使用{ \it H-检验}，则仅引入单个统计试验。这里提出的新的概率分布有利于检测较弱的信号，相对于先前已知的分布，灵敏度有所提高。
英文摘要：
Aims: To provide a significantly improved probability distribution for the H-test for periodicity in X-ray and $γ$-ray arrival times, which is already extensively used by the $γ$-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 $γ$-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 $λ=0.4$ so that the cumulative probability distribution $P(>H)=\exp{(-λH)}$. If we stack independent values for $H$, the sum of $K$ such values would follow the Erlang-K distribution with parameter $λ$ 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.
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