Q&A

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

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I was reading a paper (Kuechel et al. 2020, since retracted; see original version) 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. As such, the preprint has been removed. ↩︎

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