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

TL;DR

This paper introduces a Stein-based estimator for the intensity of a homogeneous Poisson point process that outperforms the traditional maximum likelihood estimator, especially in practical scenarios, by leveraging a new integration by parts formula.

Contribution

The paper develops a novel Stein estimator for Poisson process intensity using a new integration by parts formula, improving estimation accuracy over MLE.

Findings

Stein estimator reduces mean squared error compared to MLE

Gain in accuracy can exceed 30% in practical cases

New integration by parts formula for Poisson processes

Abstract

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 in $\R^d$ and observed in a bounded window. The procedure is based on a new general 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 particular, we show that in many practical situations we have a gain larger than 30\%.