On Estimating the Asymptotic Variance of Stationary Point Processes

Lothar Heinrich () and Michaela Prokešová ()
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Lothar Heinrich: University of Augsburg
Michaela Prokešová: Charles University of Prague

Methodology and Computing in Applied Probability, 2010, vol. 12, issue 3, 451-471

Abstract: Abstract We investigate a class of kernel estimators $\widehat{\sigma}^2_n$ of the asymptotic variance σ 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 $W_n = [-n,n]^d\,$ . σ 2 is defined by the asymptotic relation $Var(\Psi(W_n)) \sim \sigma^2 \,(2n)^d$ (as n → ∞) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{red}(\cdot)$ has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of $\gamma^{(2)}_{red}(\cdot)$ 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 $\gamma^{(2)}_{red}(\cdot)$ 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 $\widehat{\sigma}^2_n$ . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of $\widehat{\sigma}^2_n$ for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b n .

Keywords: Reduced covariance measure; Factorial moment and cumulant measures; Poisson cluster process; Hard-core process; Kernel-type estimator; Mean squared error; Optimal bandwidth; Pair correlation function; Central limit theorem; Brillinger-mixing; Primary 60G55; 62F12; Secondary 62G05; 62G20 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s11009-008-9113-3

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