A robust heuristic estimator for the period of a Poisson intensity function
type: Item
instance of: scholarly article
title: A robust heuristic estimator for the period of a Poisson intensity function
zbMATH Open document ID: 1060.62040
DOI: 10.1023/B:MCAP.0000045090.84875.22
author: Ričardas Zitikis
author: Mark Bebbington
published in: Methodology and Computing in Applied Probability
publication date: 2005-02-11T00:00:00Z
review text: 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.
reviewed by: Viktor Ohanyan
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 2133634
zbMATH Keywords: cyclic point processes
zbMATH Keywords: experimental evaluation
zbMATH Keywords: simulation
MaRDI profile type: Publication
full work available at URL: b:mcap.0000045090.84875.22
OpenAlex ID: W1985145613
description: scientific article; zbMATH DE number 2133634