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Optimisation of linear unbiased intensity estimators for point processes

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Abstract

A general non-stationary point process whose intensity function is given up to unknown numerical factor λ is considered. As an alternative to the conventional estimator of λ based on counting the points, we consider general linear unbiased estimators of λ given by sums of weights associated with individual points. A necessary and sufficient condition for a linear, unbiased estimator for the intensity λ to have the minimum variance is determined. It is shown that there are “nearly” no other processes than Poisson and Cox for which the unweighted estimator of λ, which counts the points only, is optimal. The properties of the optimal estimator are illustrated by simulations for the Matérn cluster and the Matérn hard-core processes.

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Author information

Authors and Affiliations

  1. Mathematical Department, Pedagogical Faculty, University of South Bohemia, Jeronýmova 10, 17001, České Budějovice, Czech Republic

    Tomáš Mrkvička

  2. Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, CH-3012, Bern, Switzerland

    Ilya Molchanov

Authors
  1. Tomáš Mrkvička
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  2. Ilya Molchanov
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Additional information

This research was partially supported by Grant Agency of Czech Republic, project No. 201/03/D062.

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Mrkvička, T., Molchanov, I. Optimisation of linear unbiased intensity estimators for point processes. Ann Inst Stat Math 57, 71–81 (2005). https://doi.org/10.1007/BF02506880

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  • DOI: https://doi.org/10.1007/BF02506880

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