On Estimating the Asymptotic Variance of Stationary Point Processes

Abstract

We investigate a class of kernel estimators σ 2 n of the asymptotic variance σ 2 of a d-dimensional stationary point process Ψ = i≥1 δ X i which can be observed in a cubic sampling windowand 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 b n (up to a constant) minimizing the mean squared error of σ 2 n . The case when γ (2) red (·) has bounded support is of particular interest. Further we suggest an isotropised estimator σ 2 n suitable for motion-invariant point processes and compare its properties with σ 2 . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of σ 2 for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b n .

Intensity Estimation of Stationary Fibre Processes From Digital Images With a Learned Detector

“…Regarding the estimator described in Eq. 6, since in the experiments in Heinrich and Prokešová (2010) the optimal bandwidth b W was found such that d := b w ν 2 (W ) ∈ [1, 3], we chose b W such that either d = 2 or d = 3. In fact, due to the pixel approximation, we can choose only integer values for d (in Eq.…”

“…5 we need to count the pairs of intersections (x, y) such that |x − y| <= d) and d = 1 provides too narrow neighborhood to detect pairs of intersections. We computed the mean square error (MSE) and the relative mean square error (relMSE), as in Heinrich and Prokešová (2010), to compare the quality of the estimation. For the relMSE, we used as reference (true) variance of the estimator the sample variance of the estimator calculated over the 100 independent images of the specific pair (Φ 1 , Φ 2 ).…”

“…In this situation, we need an accurate way to approximate the variance of the point process of the intersections on a single image. Heinrich and Prokešová (2010) proved that if P is a stationary point process, with milder mixing conditions than the ones required for the asymptotic normality of the corresponding counting measure, and if {W n } n is a sequence of enlarging bounded convex windows of observation, then…”

“…On simulations with small windows of observation, and different types of point process models, kernels and bandwidths, Heinrich and Prokešová (2010) obtained that the estimator, which achieved the lowest relative mean square error (E[(σ 2 P −σ 2 P ) 2 ]/σ 4 P ), was the one with the cylinder kernel with a value of b w ν 2 (W ) in the interval [1,3]. Therefore, we can derive an estimator for Var( L A,1 ) from Eqs.…”

Standard and robust intensity parameter estimation for stationary determinantal point processes

“…Heinrich and Prokešová (2010) obtained in particular the following result which can be directly applied to DPPs with kernel C satisfying A C .…”