On More Sensitive Periodogram Statistics

Period searches in event data have traditionally used the Rayleigh statistic, $R^2$. For X-ray pulsars, the standard has been the $Z^2$ statistic, which sums over more than one harmonic. For $\gamma$-rays, the $H$-test, which optimizes the number of harmonics to sum, is often used. These periodograms all suffer from the same problem, namely artefacts caused by correlations in the Fourier components that arise from testing frequencies with a non-integer number of cycles. This article addresses this problem. The modified Rayleigh statistic is discussed, its generalization to any harmonic, $\mathcal{R}^2_k$, is formulated, and from the latter, the modified $Z^2$ statistic, $\mathcal{Z}^2$, is constructed. Versions of these statistics for binned data and point measurements are derived, and it is shown that the variance in the uncertainties can have an important influence on the periodogram. It is shown how to combine the information about the signal frequency from the different harmonics to estimate its value with maximum accuracy. The methods are applied to an $\textit{XMM-Newton}$ observation of the Crab pulsar for which a decomposition of the pulse profile is presented, and shows that most of the power is in the second, third, and fifth harmonics. The statistical detection power of the $\mathcal{R}^2_k$ statistic is superior to the FFT and equivalent to the Lomb-Scargle (LS). Response to gaps in the data is assessed, and it is shown that the LS does not protect against the distortions they cause. The main conclusion of this work is that the classical $R^2$ and $Z^2$ should be replaced by $\mathcal{R}^2_k$ and $\mathcal{Z}^2$ in all applications with event data, and the LS should be replaced by the $\mathcal{R}^2_k$ when the uncertainty varies from one point measurement to another.