When and how should the $Z_n^2$ statistic be used?
I was reading a paper (Kuechel et al. 2020, since retracted; see original version) claiming a detection of a highfrequency 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 $$\phi_i=\nu(t_it_0)+\dot{\nu}(t_it_0)^2/2+\cdots$$ 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?

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 wellknown 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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By choosing $n=1$, Kuechel et al. actually use the special case of the Rayleigh statistic $R^2$, which only looks at the fundamental harmonic. In general, however, the $Z_n^2$ statistic is ideal for searching for nonsinusoidal periodic signals, which requires looking at contributions from higher harmonics and hence typically choosing $n>1$ (Belanger 2017, Buccheri 1983). The ability to look beyond the fundamental can be quite handy when looking at, among other things, the distinctly nonsinusoidal xray or gamma ray pulses from pulsars.
If you know a priori what your signal should look like, you maybe able to determine how many harmonics to use without much trouble. In this context within xray astronomy, $n=2$ may actually maximize the signaltonoise ratio of an observation, as it provides sensitivity to unknown pulses with a range of different broadnesses. An optimal number of harmonics, however, may be determined by applying the $H$test of de Jager et al. 1989. Here, we define the $H$statistic by $$H\equiv\max_{1\leq n\leq20}(Z_n^24n+4)$$ Realistically, an infinite number of harmonics could be searched, but the authors argue that $n\leq20$ is typically an adequate truncation, at least for these purposes. $H$ itself can also in fact be used as a statistical test in place of $Z_n^2$.
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