Nonparametric Testing of Distribution Functions in Germ-grain Models

  • Zbyněk Pawlas
  • Lothar Heinrich
Part of the Lecture Notes in Statistics book series (LNS, volume 185)


Germ-grain models are random closed sets in the d-dimensional Euclidean space ℝd which admit a representation as union of random compact sets (called grains) shifted by the atoms (called germs) of a point process. In this note we consider the distribution function F of an m-dimensional random vector describing shape and size parameters of the typical grain of a stationary germ-grain model. We suggest a ratio-unbiased weighted (Horvitz-Thompson type) empirical distribution function \(\hat F_n \) to estimate F, based on the corresponding data vectors of those shifted grains which lie completely within the sampling window Wn ⊆ ℝd. Since, as Wn increases, the empirical process \(\hat F_n \)(t) − F(t) (after scaling) converges weakly to an m-parameter Brownian bridge process, it is possible for the particular case where m = 1, to examine the the goodness-of-fit of observed data to a hypothesised continuous distribution function F, analogous to the Kolmogorov-Smirnov test.

Key words

Germ-grain model Horvitz-Thompson-type estimator Kolmogorov-Smirnov test Multivariate empirical process Weak convergence 


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  1. [1]
    P. Billingsley. Convergence of Probability Measures. 2nd Edition, Wiley & Sons, New York, 1999.Google Scholar
  2. [2]
    D.J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes. Springer-Verlag, New York, 1988.Google Scholar
  3. [3]
    L. Heinrich and Z. Pawlas. Weak and strong convergence of empirical distribution functions in germ-grain models. Submitted, 2004.Google Scholar
  4. [4]
    B.G. Ivanoff. The function space D([0,∞)q,E). Canadian Journal of Statistics, 8:173–186, 1980.MathSciNetGoogle Scholar
  5. [5]
    R. Júzková, P. Ctibor and V. Beneš. Analysis of porous structure in plasma-sprayed coating. Image Analysis and Stereology, 23:45–52, 2004.Google Scholar
  6. [6]
    G. Neuhaus. On the weak convergence of stochastic processes with multidimensional time parameter. Annals of Mathematical Statistics, 42:1285–1295, 1971.zbMATHMathSciNetGoogle Scholar
  7. [7]
    R. Schneider and W. Weil. Stochastische Geometrie. B.G. Teubner, Stuttgart, Leipzig, 2000.Google Scholar
  8. [8]
    D. Stoyan, W.S. Kendall and J. Mecke. Stochastic Geometry and its Applications. 2nd Edition, Wiley & Sons, Chichester, 1995.Google Scholar

Copyright information

© Springer Science+Business Media Inc. 2006

Authors and Affiliations

  • Zbyněk Pawlas
    • 1
    • 2
  • Lothar Heinrich
    • 3
  1. 1.Faculty of Mathematics and Physics, Department of Probability and Mathematical StatisticsCharles UniversityPraha 8Czech Republic
  2. 2.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPraha 8Czech Republic
  3. 3.Institute of MathematicsUniversity of AugsburgAugsburgGermany

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