On Estimating the Asymptotic Variance of Stationary Point Processes
On Estimating the Asymptotic Variance of Stationary Point Processes
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DOI：
10.1007/s11009-008-9113-3
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发表时间：
2010-09-01
影响因子：
0.9
通讯作者：
Prokesova, Michaela
中科院分区：
文献类型：
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作者：
Heinrich, Lothar;Prokesova, Michaela
我们研究了d维平稳点过程Psi = sigma(i >= 1) delta（Xi）的渐近方差sigma(2)的cap (2)(n)上的一类核估计（sigma），它可以在三次采样窗口W(n) =[-n, n](d)中观察到。sigma 2由类似于sigma(2) (2n)(d) （n ->∞）的渐近关系Var(Psi（W(n)）定义，当对应的约化协方差测量时，保证其存在。Gamma((2))（红色）（.）的总变化是有限的。根据r(ed)((2))（.）在以原点为中心的膨胀球外的总变化的衰减率（多项式或指数），我们确定最佳带宽b(n)（直到一个常数）最小化（sigma）超过(2)(n)的均方误差。这种情况下。R (ed)((2))（.）具有有限支持是特别有趣的。此外，我们提出了一个适用于运动不变点过程的各向同性估计量（类似于）(2)(sigma)(n)，并将其性质与(sigma) / cap (2)(n)进行比较。我们的理论结果得到了模拟研究的说明和支持，该研究比较了平面泊松、泊松聚类和硬核点过程以及不同nb(n)值的（相对）均方误差（sigma）在cap (2)(n)上的（相对）均方误差。
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.)