Gaussian Limits of Empirical Multiparameter K-Functions of Homogeneous Poisson Processes and Tests for Complete Spatial Randomness

  • We prove two functional limit theorems for empirical multiparameter second moment functions (generalizing Ripley's K-function) obtained from a homogeneous Poisson point field observed in an unboundedly expanding convex sampling window W_n in R^d. The cases of known and unknown (estimated) intensity lead to distinct Gaussian limits and require quite different proofs. Further we determine the limit distributions of the maximal deviation and the integrated squared distance between empirical and true multiparameter second moment function. These results give rise to construct goodness-of-fit tests for checking the hypothesis that a given point pattern is completely spatially random (CSR), i.e. a realization of a homogeneous Poisson process.

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Author:Lothar HeinrichGND
Frontdoor URL
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2014-06)
Publishing Institution:Universität Augsburg
Release Date:2014/10/06
Tag:point process; reduced second moment measure; set-indexed Gaussian process; multiparameter Gaussian process; Wiener sheet; m-dependence; U-statistic; weak convergence; Skorokhod-space of multiparameter cadlag-functions; goodness-of-fit tests
GND-Keyword:Punktprozess; Güte der Anpassung; Gauß-Prozess; U-Statistik; Schwache Konvergenz
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Stochastik und ihre Anwendungen
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand