When and how should the $Z_n^2$ statistic be used?


I was reading a paper (Kuechel et al. 2020) claiming a detection of a high-frequency periodic signal coming from a known pulsar.1 The authors used something called the $Z_n^2$ statistic, a test I'm not familiar with. Assuming we have a time series of $N+1$ photons with arrival times $t_0,t_1,t_2,\dots,t_N$, we write the phase of the $i$th photon after the first as $$\phi_i=\nu(t_i-t_0)+\dot{\nu}(t_i-t_0)^2/2+\cdots$$ with $\nu,\dot{\nu},\dots$ being the frequency and frequency time derivatives of the signal. The $Z_n^2$ statistic is then defined as $$Z_n^2=\frac{2}{N}\sum_{k=1}^n\left[\left(\sum_{i=1}^N\cos(2\pi k\phi_i)\right)^2+\left(\sum_{i=1}^N\sin(2\pi k\phi_i)\right)^2\right]$$ Higher values of $Z_n^2$ appear to correspond to a strong statistical significance of a signal. In this case, the authors picked $n=1$ and used the $Z_1^2$ statistic.

I've never encountered the $Z_n^2$ statistic before, and I've had a hard time tracking down and/or accessing references, so my question has to do with its applicability. I assume it should be used only for periodic signals - is that right? If so, what values of $n$ are appropriate for these signals? Is $n=1$ usually adequate?

  1. As a side note, folks who know more about the instrument used than I do are fairly confident that this detection isn't actually coming from the system. While the source has a known frequency of $\sim567$ Hz, the frequency Kuechel et al. report detecting is $\sim890$ Hz, which corresponds to a well-known instrumental frequency of the telescope, NuSTAR. Therefore, while the signal is real, it's likely not astronomical in origin.

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Worth mentioning that the paper has now been taken down as they were detecting "dead time in the NuSTAR detectors". This is apparently "further evidence for why standard timing methods should not be used with NuSTAR data" ‭Mithrandir24601‭ about 11 hours ago

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