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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 $\widehat{\sigma}^2_n$ of the asymptotic variance $\sigma^2$ of a $d$--dimensional stationary point process $\Psi = \sum_{i\ge 1}\delta_{X_i}$ which can be observed in a cubic sampling window $W_n = [-n,n]^d\,$. $\sigma^2$ is defined by the asymptotic relation $\mathsf{Var}(\Psi(W_n)) \sim \sigma^2 \,(2n)^d$ (as $n \to \infty$) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{\mathrm{red}}(\cdot)$ has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of $\gamma^{(2)}_{\mathrm{red}}(\cdot)$ outside of an expanding ball centered at the origin, we determine optimal bandwidths $b_n$ (up to a constant) minimizing the mean squared error of $\widehat{\sigma}^2_n$. The case when $\gamma^{(2)}_{\mathrm{red}}(\cdot)$ has bounded support is of particular interest. Further we suggest an isotropised estimator $\widetilde{\sigma}^2_n$ suitable for motion-invariant point processes and compare its properties with $\widehat{\sigma}^2$. Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of $\widehat{\sigma}^2$ for planar Poisson, Poisson cluster, and hard--core point processes and for various values of $n\,b_n\,$.
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This primarily serves as Thiele Research Reports number 17-2006, but was also published in Research Reports