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reduced covariance measure; factorial moment and cumulant measures; kernel-type estimator; subsampling; mean squared error; Poisson cluster process; hard-core process
We investigate estimators of the asymptotic variance $\sigma^2$ of a $d$–dimensional stationary point process $\Psi$ which can be observed in convex and compact sampling window $W_n=n\, W$. Asymptotic variance of $\Psi$ is defined by the asymptotic relation ${Var}(\Psi(W_n)) \sim \sigma^2 |W_n|$ (as $n \to \infty$) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{{\rm red}}(\cdot)$ 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.
[1] S. Böhm, L. Heinrich, V. Schmidt: Kernel estimation of the spectral density of stationary random closed sets. Austral. & New Zealand J. Statist. 46 (2004), 41-52. DOI 10.1111/j.1467-842X.2004.00310.x | MR 2060951 | Zbl 1061.62057
[2] K. L. Chung: A Course in Probability Theory. Second edition. Harcourt Brace Jovanovich, New York 1974. MR 0346858 | Zbl 0345.60003
[3] D. J. Daley, D. Vere-Jones: An Introduction to the Theory of Point Processes. Second edition. Vol I and II, Springer, New York 2003, 2008. MR 0950166 | Zbl 1026.60061
[4] S. David: Central Limit Theorems for Empirical Product Densities of Stationary Point Processes. Phd. Thesis, Augsburg Universität 2008.
[5] L. Heinrich: Asymptotic gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes. Statistics 19 (1988), 87-106. DOI 10.1080/02331888808802075 | MR 0921628 | Zbl 0666.62032
[6] L. Heinrich: Normal approximation for some mean-value estimates of absolutely regular tesselations. Math. Methods Statist. 3 (1994), 1-24. MR 1272628
[7] L. Heinrich: Asymptotic goodness-of-fit tests for point processes based on scaled empirical K-functions. Submitted.
[8] L. Heinrich, E. Liebscher: Strong convergence of kernel estimators for product densities of absolutely regular point processes. J. Nonparametric Statist. 8 (1997), 65-96. DOI 10.1080/10485259708832715 | MR 1658113 | Zbl 0884.60041
[9] L. Heinrich, M. Prokešová: On estimating the asymptotic variance of stationary point processes. Methodology Comput. in Appl. Probab. 12 (2010), 451-471. DOI 10.1007/s11009-008-9113-3 | MR 2665270 | Zbl 1197.62122
[10] L. Heinrich, V. Schmidt: Normal convergence of multidimensional shot noise and rates of this convergence. Adv. in Appl. Probab. 17 (1985), 709-730. DOI 10.2307/1427084 | MR 0809427 | Zbl 0609.60036
[11] J. Illian, A. Penttinen, H. Stoyan, D. Stoyan: Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley and Sons, Chichester 2008. MR 2384630 | Zbl 1197.62135
[12] E. Jolivet: Central limit theorem and convergence of empirical processes of stationary point processes. In: Point Processes and Queueing Problems (P. Bartfai and J. Tomko, eds.), North-Holland, New York 1980, pp. 117-161. MR 0617406
[13] S. Mase: Asymptotic properties of stereological estimators for stationary random sets. J. Appl. Probab. 19 (1982), 111-126. DOI 10.2307/3213921 | MR 0644424
[14] D. N. Politis: Subsampling. Springer, New York 1999. MR 1707286 | Zbl 1072.62551
[15] D. N. Politis, M. Sherman: Moment estimation for statistics from marked point processes. J. Roy. Statist. Soc. Ser. B 63 (2001), 261-275. DOI 10.1111/1467-9868.00284 | MR 1841414 | Zbl 0979.62074
[16] M. Prokešová, E. B. Vedel-Jensen: Asymptotic Palm likelihood theory for stationary point processes. Submitted.
[17] B. D. Ripley: Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge 1988. MR 0971986 | Zbl 0716.62100
[18] D. Stoyan, W. S. Kendall, J. Mecke: Stochastic Geometry and its Applications. Second edition. J. Wiley & Sons, Chichester 1995. MR 0895588 | Zbl 0838.60002
[19] J. C. Taylor: An Introduction to Measure and Probability. Springer, New York 1997. MR 1420194
[20] L. Zhengyan, L. Chuanrong: Limit Theory for Mixing Dependent Random Variables. Kluwer Academic Publishers, Dordrecht 1996. MR 1486580 | Zbl 0889.60001
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