  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.
