Bernoulli

  • Bernoulli
  • Volume 20, Number 4 (2014), 1673-1697.

Asymptotic goodness-of-fit tests for the Palm mark distribution of stationary point processes with correlated marks

Lothar Heinrich, Sebastian Lück, and Volker Schmidt

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Abstract

We consider spatially homogeneous marked point patterns in an unboundedly expanding convex sampling window. Our main objective is to identify the distribution of the typical mark by constructing an asymptotic $\chi^{2}$-goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a smoothed covariance estimator which turns out to be mean square consistent. Our approach does not require independent marks and allows dependences between the mark field and the point pattern. Instead we impose a suitable $\beta$-mixing condition on the underlying stationary marked point process which can be checked for a number of Poisson-based models and, in particular, in the case of geostatistical marking. In order to study test performance, our test approach is applied to detect anisotropy of specific Boolean models.

Article information

Source
Bernoulli, Volume 20, Number 4 (2014), 1673-1697.

Dates
First available in Project Euclid: 19 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1411134440

Digital Object Identifier
doi:10.3150/13-BEJ523

Mathematical Reviews number (MathSciNet)
MR3263085

Zentralblatt MATH identifier
1312.60062

Keywords
$\beta$-mixing point process empirical Palm mark distribution reduced factorial moment measures smoothed covariance estimation $\chi^{2}$-goodness-of-fit test

Citation

Heinrich, Lothar; Lück, Sebastian; Schmidt, Volker. Asymptotic goodness-of-fit tests for the Palm mark distribution of stationary point processes with correlated marks. Bernoulli 20 (2014), no. 4, 1673--1697. doi:10.3150/13-BEJ523. https://projecteuclid.org/euclid.bj/1411134440


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References

  • [1] Beneš, V., Hlawiczková, M., Gokhale, A. and Vander Voort, G. (2001). Anisotropy estimation properties for microstructural models. Mater. Charact. 46 93–98.
  • [2] Böhm, S. and Schmidt, V. (2004). Asymptotic properties of estimators for the volume fractions of jointly stationary random sets. Stat. Neerl. 58 388–406.
  • [3] Bradley, R.C. (2007). An Introduction to Strong Mixing Conditions. Vols 1, 2, 3. Heber City, UT: Kendrick Press.
  • [4] Daley, D. and Vere-Jones, D. (2003/2008). An Introduction to the Theory of Point Processes. Vols I, II, 2nd ed. New York: Springer.
  • [5] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. New York: Springer.
  • [6] Folland, G.B. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Pure and Applied Mathematics (New York). New York: Wiley.
  • [7] Guan, Y., Sherman, M. and Calvin, J.A. (2004). A nonparametric test for spatial isotropy using subsampling. J. Amer. Statist. Assoc. 99 810–821.
  • [8] Guan, Y., Sherman, M. and Calvin, J.A. (2007). On asymptotic properties of the mark variogram estimator of a marked point process. J. Statist. Plann. Inference 137 148–161.
  • [9] Heinrich, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations. Math. Methods Statist. 3 1–24.
  • [10] Heinrich, L., Klein, S. and Moser, M. (2014). Empirical mark covariance and product density function of stationary marked point processes – A survey on asymptotic results. Methodol. Comput. Appl. Probab. 16 283–293.
  • [11] Heinrich, L., Lück, S., Nolde, M. and Schmidt, V. (2014). On strong mixing, Bernstein’s blocking method and a CLT for spatial marked point processes. Yokohama Math. J. To appear.
  • [12] Heinrich, L., Lück, S. and Schmidt, V. (2012). Non-parametric asymptotic statistics for the Palm mark distribution of $\beta $-mixing marked point processes. Available at arXiv:1205.5044v1 [math.ST].
  • [13] Heinrich, L. and Molchanov, I.S. (1999). Central limit theorem for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. 31 283–314.
  • [14] Heinrich, L. and Pawlas, Z. (2008). Weak and strong convergence of empirical distribution functions from germ-grain processes. Statistics 42 49–65.
  • [15] Heinrich, L. and Prokešová, M. (2010). On estimating the asymptotic variance of stationary point processes. Methodol. Comput. Appl. Probab. 12 451–471.
  • [16] Kallenberg, O. (1986). Random Measures. London: Academic Press.
  • [17] Lück, S., Kupsch, A., Lange, A., Hentschel, M. and Schmidt, V. (2012). Statistical analysis of tomographic reconstruction algorithms by morphological image characteristics. Mater. Res. Soc. Symp. Proc. 1421. DOI:10.1557/opl.2012.209.
  • [18] Midgley, P.A. and Weyland, M. (2003). 3D electron microscopy in the physical sciences: The development of Z-contrast and EFTEM tomography. Ultramicroscopy 96 413–431.
  • [19] Pawlas, Z. (2009). Empirical distributions in marked point processes. Stochastic Process. Appl. 119 4194–4209.
  • [20] Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications 44. Cambridge: Cambridge Univ. Press.
  • [21] Yoshihara, K.I. (1976). Limiting behavior of $U$-statistics for stationary, absolutely regular processes. Z. Wahrsch. Verw. Gebiete 35 237–252.