  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.
