A robust heuristic estimator for the period of a Poisson intensity function

While there are a number of methods for estimating the intensity of a cyclic point process, these assume prior estimation of the period itself. The standard method for the latter is the "periodogram," or spectral analysis, approach. This is a parametric method which is sensitive to the form, in particular the number of peaks per cycle, of the intensity. We construct a family of nonparametric estimators for the period of a cyclic Poisson process, with the object of robustness against the form of the intensity. These are tested, along with the standard periodogram estimate and an earlier nonparametric estimator, on simulated data from a range of intensity functions. While the nonparametric estimators presently lack the well-developed asymptotic and statistical properties of the periodogram, the best of them is almost as accurate as the periodogram for the unimodal intensity cycles on which the latter is based. Whereas the periodogram cannot handle multimodal cycles at all, the better nonparametric estimators are reasonably accurate, and sometimes err by estimating multiples of the period rather than divisors, errors that are arguably less damaging. We conclude with some remarks concerning the derivation of asymptotic properties for our nonparametric estimator.