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Two-step estimation procedures for inhomogeneous shot-noise Cox processes

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Abstract

In the present paper, we discuss and compare several two-step estimation procedures for inhomogeneous shot-noise Cox processes. The intensity function is parametrized by the inhomogeneity parameters while the pair-correlation function is parametrized by the interaction parameters. The suggested procedures are based on a combination of Poisson likelihood estimation of the inhomogeneity parameters in the first step and an adaptation of a method from the homogeneous case for estimation of the interaction parameters in the second step. The adapted methods, based on minimum contrast estimation, composite likelihood and Palm likelihood, are compared both theoretically and by means of a simulation study. The general conclusion from the simulation study is that the three estimation methods have similar performance. Two-step estimation with Palm likelihood has not been considered before and is motivated by the superior performance of the Palm likelihood in the stationary case for estimation of certain parameters of interest. Asymptotic normality of the two-step estimator with Palm likelihood is proved.

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Acknowledgments

This project has been supported by the Czech Science Foundation, Project No. P201/10/0472, Charles University Grant Agency, Project No. 664313, and by Centre for Stochastic Geometry and Advanced Bioimaging, funded by a Grant from The Villum Foundation. We thank the referees for constructive comments and suggestions both concerning the content and exposition of the paper.

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Authors and Affiliations

  1. Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, Praha 8, 186 75, Czech Republic

    Michaela Prokešová & Jiří Dvořák

  2. Department of Mathematics, Centre for Stochastic Geometry and Advanced Bioimaging, Aarhus University, Ny Munkegade 118, Bldg. 1530, 8000, Aarhus C, Denmark

    Eva B. Vedel Jensen

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  1. Michaela Prokešová
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  2. Jiří Dvořák
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  3. Eva B. Vedel Jensen
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Correspondence to Michaela Prokešová.

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Prokešová, M., Dvořák, J. & Jensen, E.B.V. Two-step estimation procedures for inhomogeneous shot-noise Cox processes. Ann Inst Stat Math 69, 513–542 (2017). https://doi.org/10.1007/s10463-016-0556-y

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  • DOI: https://doi.org/10.1007/s10463-016-0556-y

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