On Estimating the Asymptotic Variance of Stationary Point Processes

We investigate a class of kernel estimators (sigma) over cap (2)(n) of the asymptotic variance sigma(2) of a d-dimensional stationary point process Psi = Sigma(i >= 1) delta(Xi) which can be observed in a cubic sampling window W(n) =[-n, n](d). sigma 2 is defined by the asymptotic relation Var(Psi(W(n)))similar to sigma(2) (2n)(d) (as n -> infinity) and its existence is guaranteed whenever the corresponding reduced covariance measure. gamma((2))(red) (.) has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of r(ed)((2))(.) 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 (sigma) over cap (2)(n). The case when. r(ed)((2))(.) has bounded support is of particular interest. Further we suggest an isotropised estimator (similar to)(2)(sigma)(n) suitable for motion-invariant point processes and compare its properties with (sigma) over cap (2)(n). Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of (sigma) over cap (2)(n) for planar Poisson, Poisson cluster, and hard-core point processes and for various values of nb(n.)