Nonparametric Testing of Distribution Functions in Germ-grain Models
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 wordsGerm-grain model Horvitz-Thompson-type estimator Kolmogorov-Smirnov test Multivariate empirical process Weak convergence
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