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Boolean model; Boolean model hypothesis; contact distribution function; Euler–Poincaré characteristic; Intrinsic volumes; Laslettś transform
A new method of testing the random closed set model hypothesis (for example: the Boolean model hypothesis) for a stationary random closed set $\Xi\subseteq{{\mathbb R}^d}$ with values in the extended convex ring is introduced. The method is based on the summary statistics – normalized intrinsic volumes densities of the $\varepsilon$-parallel sets to $\Xi$. The estimated summary statistics are compared with theirs envelopes produced from simulations of the model given by the tested hypothesis. The p-level of the test is then computed via approximation of the summary statistics by multinormal distribution which mean and the correlation matrix is computed via given simulations. A new estimator of the intrinsic volumes densities from [6] is used, which is especially suitable for estimation of the intrinsic volumes densities of $\varepsilon$-parallel sets. The power of this test is estimated for planar Boolean model hypothesis and two different alternatives and the resulted powers are compared to the powers of known Boolean model tests. The method is applied on the real data set of a heather incidence.
[1] N. A. C. Cressie: Statistics for Spatial Data. Revised Edition. Wiley, New York 1997. MR 1127423 | Zbl 0799.62002
[2] P. J. Diggle: Binary mosaics and the spatial pattern of heather. Biometrics 37 (1981), 531–539.
[3] I. Molchanov: Statistics of the Boolean Model for Practitionars and Mathematicians. Wiley, New York 1997.
[4] J. Møller and K. Helisová: Power diagrams and interaction processes for union of discs. Adv. in Appl. Probab. (SGSA) 40 (2008), 1–27. MR 2431299
[5] J. Møller and R. P. Waagepetersen: Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, London 2004 MR 2004226
[6] T. Mrkvička and J. Rataj: On estimation of intrinsic volume densities of stationary random closed sets. Stochastic Process. Appl. 118/2 (2008), 213–231. MR 2376900
[7] W. Nagel, J. Ohser, and K. Pischang: An integral-geometric approach for the Euler–Poincaré characteristic of spatial images. J. Microsc. 198 (2000), 54–62.
[8] Ohser J. and F. Mücklich: Statistical Analysis of Microstructures in Materials Science. Wiley, Chichester 2000.
[9] J. Rataj: On estimation of the Euler number by projections of thin slabs. Adv. in Appl. Probab. 36 (2004), 715–724. MR 2079910 | Zbl 1070.60010
[10] J. Rataj: Estimation of intrinsic volumes from parallel neighbourhoods. Suppl. Rend. Circ. Mat. Palermo, Ser. II 77 (2006), 553-563. MR 2245722 | Zbl 1101.62084
[11] V. Schmidt and E. Spodarev: Joint estimators for the specific intrinsic volumes of stationary random sets. Stochastic Process. Appl. 115 (2005), 959–981. MR 2138810
[12] R. Schneider: Convex Bodies. The Brunn–Minkowski Theory. Cambridge Univ. Press, Cambridge 1993. MR 1216521 | Zbl 1168.52002
[13] R. Schneider and W. Weil: Stochastische Geometrie. Teubner, Stuttgart 2000. MR 1794753
[14] D. Stoyan, W. S. Kendall, and J. Mecke: Stochastic Geometry and Its Applications. Second edition. Wiley, Chichester 1995. MR 0895588
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