The Annals of Applied Probability

Stein estimation of the intensity of a spatial homogeneous Poisson point process

Marianne Clausel, Jean-François Coeurjolly, and Jérôme Lelong

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In this paper, we revisit the original ideas of Stein and propose an estimator of the intensity parameter of a homogeneous Poisson point process defined on $\mathbb{R}^{d}$ and observed on a bounded window. The procedure is based on a new integration by parts formula for Poisson point processes. We show that our Stein estimator outperforms the maximum likelihood estimator in terms of mean squared error. In many practical situations, we obtain a gain larger than 30%.

Article information

Ann. Appl. Probab., Volume 26, Number 3 (2016), 1495-1534.

Received: July 2014
Revised: March 2015
First available in Project Euclid: 14 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Stein formula Malliavin calculus superefficient estimator intensity estimation spatial point process


Clausel, Marianne; Coeurjolly, Jean-François; Lelong, Jérôme. Stein estimation of the intensity of a spatial homogeneous Poisson point process. Ann. Appl. Probab. 26 (2016), no. 3, 1495--1534. doi:10.1214/15-AAP1124.

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