A robust heuristic estimator for the period of a Poisson intensity function (Q1762884)

Methods for analyzing naturally periodic, or cyclic, data are well understood when the data form a time-series. However, often the data consist of events randomly located in the time span. Point processes form a natural stochastic model for such data. Let \(X\) be a nonhomogeneous Poisson point process in the real line \(\mathbb R\) with (unknown) locally integrable intensity function \(\lambda(s)\), such that \(\lambda(s) =\lambda(s+k\,\tau)\), \(k=\pm1, \pm 2,\dots\), for some period \(\tau>0\). If a theoretical model of \(\lambda(s)\) is given in a parametric form, then one may use maximum likelihood to estimate the unknown parameters and thus obtain an estimator for \(\lambda(s)\). \textit{P. Hall, J. Reimann}, and \textit{J. Rice} [Biometrica 87, No. 3, 545--557 (2000; Zbl 0956.62031)] considered a nonparametric form of \(\lambda(s)\) in the time series case. \textit{R. Helmers, I.W. Mangku} and \textit{R. Zitikis} [J. Multivariate Anal., 84, No. 1, 19--39 (2003; Zbl 1038.62037)] proposed a completely nonparametric estimator. The above methods all have in common the fact that the period of the intensity is assumed to be known, either from prior information, or by using the standard `periodogram' method. If the period \(\tau>0\) is unknown, this implies the need for a robust estimator \(\widehat \tau\), independent of the shape of the intensity function. The construction of estimators \(\widehat \tau\) with desirable properties appears to be a challenging problem. In Section 2, the authors propose a family of nonparametric estimators, explain the underlying reasoning, and explore a few of their theoretical properties. In Section 3 (experimental evaluation), the authors trial all of the estimators on simulated data from a range of periodic intensities. The remaining part of the paper concludes a discussion concerning the derivation of asymptotic properties for the new estimators.