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On estimating the asymptotic variance of stationary point processes

by Lothar Heinrich and Michaela Prokesová
Research Reports Number 486 (October 2006)
We investigate a class of kernel estimators ˆσ2n of the asymptotic variance σ2 of a d--dimensional stationary point process Ψ=i1δXi which can be observed in a cubic sampling window Wn=[n,n]d. σ2 is defined by the asymptotic relation Var(Ψ(Wn))σ2(2n)d (as n) and its existence is guaranteed whenever the corresponding reduced covariance measure γ(2)red() has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of γ(2)red() outside of an expanding ball centered at the origin, we determine optimal bandwidths bn (up to a constant) minimizing the mean squared error of ˆσ2n. The case when γ(2)red() has bounded support is of particular interest. Further we suggest an isotropised estimator ˜σ2n suitable for motion-invariant point processes and compare its properties with ˆσ2. Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of ˆσ2 for planar Poisson, Poisson cluster, and hard--core point processes and for various values of nbn.
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This primarily serves as Thiele Research Reports number 17-2006, but was also published in Research Reports