Not yet reviewed by Pith; the record is open.

The estimation of periodicity is a fundamental task in many scientific areas of study. Existing methods rely on theoretical assumptions that the observation times have equal or i.i.d. spacings, and that common estimators, such as the periodogram peak, are consistent and asymptotically normal. In practice, however, these assumptions are unrealistic as observation times usually exhibit deterministic patterns -- e.g., the nightly observation cycle in astronomy -- that imprint nuisance periodicities in the data. These nuisance signals also affect the finite-sample distribution of estimators, which can substantially deviate from normality. Here, we propose a set identification method, fusing ideas from randomization inference and partial identification. In particular, we develop a sharp test for any periodicity value, and then invert the test to build a confidence set. This approach is appropriate here because the construction of confidence sets does not rely on assumptions of regular or well-behaved asymptotics. Notably, our inference is valid in finite samples when our method is fully implemented, while it can be asymptotically valid under an approximate implementation designed to ease computation. Empirically, we validate our method in exoplanet detection using radial velocity data. In this context, our method correctly identifies the periodicity of the confirmed exoplanets in our sample. For some other, yet unconfirmed detections, we show that the statistical evidence is weak, which illustrates the failure of traditional statistical techniques. Last but not least, our method offers a constructive way to resolve these identification issues via improved observation designs. In exoplanet detection, these designs suggest meaningful improvements in identifying periodicity even when a moderate amount of randomization is introduced in scheduling radial velocity measurements.