Nonparametric Inference for Periodic Sequences

This article proposes a nonparametric method for estimating the period and values of a periodic sequence when the data are evenly spaced in time. The period is estimated by a “leave-out-one-cycle” version of cross-validation (CV) and complements the periodogram, a widely used tool for period estimation. The CV method is computationally simple and implicitly penalizes multiples of the smallest period, leading to a “virtually” consistent estimator of integer periods. This estimator is investigated both theoretically and by simulation. We also propose a nonparametric test of the null hypothesis that the data have constant mean against the alternative that the sequence of means is periodic. Finally, our methodology is demonstrated on three well-known time series: the sunspots and lynx trapping data, and the El Niño series of sea surface temperatures.