The aim of our paper is to investigate the properties of the classical phase-dispersion minimization (PDM), analysis of variance (AOV), string-length (SL), and Lomb–Scargle (LS) power statistics from a statistician’s perspective. We confirm that when the data are perturbations of a constant function, i.e. under the null hypothesis of no period in the data, a scaled version of the PDM statistic follows a beta distribution, the AOV statistic follows an F distribution, and the LS power follows a chi-squared distribution with two degrees of freedom. However, the SL statistic does not have a closed-form distribution. We further verify these theoretical distributions through simulations and demonstrate that the extreme values of these statistics (over a range of trial periods), often used for period estimation and determination of the false alarm probability (FAP), follow different distributions than those derived for a single period. We emphasize that multiple-testing considerations are needed to correctly derive FAP bounds. Though, in fact, multiple-testing controls are built into the FAP bound for these extreme-value statistics, e.g. the FAP bound derived specifically for the maximum LS power statistic over a range of trial periods. Additionally, we find that all of these methods are robust to heteroscedastic noise aimed to mimic the degradation or miscalibration of an instrument over time. Finally, we examine the ability of these statistics to detect a non-constant periodic function via simulating data that mimics a well-detached binary system, and we find that the AOV statistic has the most power to detect the correct period, which agrees with what has been observed in practice.

中文翻译：

---

天文学中经典周期查找技术的统计入门


我们论文的目的是从统计学家的角度研究经典相色散最小化 (PDM)、方差分析 (AOV)、弦长度 (SL) 和 Lomb-Scargle (LS) 功率统计的特性。我们确认，当数据是常数函数的扰动时，即在数据中没有周期的零假设下，PDM 统计量的缩放版本遵循 beta 分布，AOV 统计量遵循 F 分布，LS 幂遵循具有两个自由度的卡方分布。然而，SL 统计量没有封闭形式的分布。我们通过模拟进一步验证这些理论分布，并证明这些统计数据的极值（在一系列试验期内）通常用于周期估计和误报概率（FAP）的确定，遵循与从单期。我们强调需要考虑多重测试才能正确推导 FAP 界限。事实上，针对这些极值统计数据，FAP 界限中内置了多重测试控制，例如FAP 界限专门针对一系列试验期间的最大 LS 功率统计而得出。此外，我们发现所有这些方法对于旨在模拟仪器随时间的退化或误校准的异方差噪声都很稳健。最后，我们通过模拟独立二元系统的模拟数据来检查这些统计量检测非常量周期函数的能力，我们发现 AOV 统计量最有能力检测正确的周期，这与已在实践中观察到。