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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)

Summary

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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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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