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Testing Equality of Distributions of Random Convex Compact Sets via Theory of \(\mathfrak {N}\)-Distances

  • Vesna Gotovac DogašEmail author
  • Kateřina Helisová
Article
  • 6 Downloads

Abstract

This paper concerns a method of testing the equality of distributions of random convex compact sets. The main theoretical result involves a construction of a metric on the space of distributions of random convex compact sets. We obtain it by using the theory of \(\mathfrak {N}\)-distances and the redefined characteristic function of random convex compact set. We propose an approximation of the metric through its finite-dimensional counterparts. This result leads to a new statistical test for testing the equality of distributions of two random convex compact sets. Consequently, we show a heuristic approach how to determine whether two realisations of random sets that can be approximated by a union of identically distributed random convex compact sets come from the same underlying process using the constructed test. Each procedure is justified by an extensive simulation study and the heuristic method for comparing random sets using their convex compact counterparts is moreover applied to real data concerning histological images of two different types of mammary tissue.

Keywords

Characteristic function Non-parametric method Permutation test Support function Two-sample problem 

Mathematics Subject Classification (2010)

62G10 60D05 

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Notes

Acknowledgements

The research was supported by The Czech Science Foundation,project No. 19-04412S.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Science, Department of MathematicsUniversity of SplitSplitCroatia
  2. 2.Faculty of Electrical Engineering, Department of MathematicsCzech Technical University in PraguePrague 6Czech Republic

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