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=∑i≥1δXi 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.
This primarily serves as Thiele Research Reports number 17-2006, but was also published in Research Reports