Kybernetika 47 no. 5, 678-695, 2011

Estimators of the asymptotic variance of stationary point processes - a comparison

Michaela Prokešová


We investigate estimators of the asymptotic variance <span class="tex">\ss</span> of a <span class="tex">d</span>-dimensional stationary point process <span class="tex">&#936;</span> which can be observed in convex and compact sampling window <span class="tex"><sub>n</sub>=n W</span>. Asymptotic variance of <span class="tex">&#936;</span> is defined by the asymptotic relation <span class="tex">{Var}(&#936;(<sub>n</sub>)) \sim ß|<sub>n</sub>|</span> (as <span class="tex">n \to &#8734;</span>) and its existence is guaranteed whenever the corresponding reduced covariance measure <span class="tex">\gamr(\cdot)</span> has finite total variation. The three estimators discussed in the paper are the kernel estimator, the estimator based on the second order intesity of the point process and the subsampling estimator. We study the mean square consistency of the estimators. Since the expressions for the variance of the estimators are not available in closed form and depend on higher order moment measures of the point process, only the bias of the estimators can be compared theoretically. The second part of the paper is therefore devoted to a simulation study which compares the efficiency of the estimators by means of the mean squared error and for several clustered and repulsive point processes observed on middle-sized windows.


reduced covariance measure, factorial moment and cumulant measures, kernel-type estimator, subsampling, mean squared error, Poisson cluster process, hard-core process


60G55, 62F12


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