Some statistical remarks on the Giant GRB Ring
We studied some statistical properties of the spatial point process displayed by GRBs of known redshift. To find ring like point patterns we developed an algorithm and defined parameters to characterize the level of compactness and regularity of the rings found in this procedure. Applying this algorithm to the GRB sample we identified three more ring like point patterns. Although, they had the same regularity but much less level of compactness than the original GRB ring. Assuming a stochastic independence of the angular and radial positions of the GRBs we obtained 1502 additional samples, altogether 542222 data points, by bootstrapping the original one. None of these data points participated in rings having similar level of compactness and regularity as the original one. Using an appropriate kernel we estimated the joint probability density of the angular and radial variables of the GRBs. Performing MCMC simulations we obtained 1502 new samples, altogether 542222 data points. Among these data points only three represented ring like patterns having similar parameters as the original one. By defining a new statistical variable we tested the independence of the angular and radial variables of the GRBs. We concluded that despite the existence of local irregularities in the GRBs' spatial distribution (e.g. the GGR) one can not reject the Cosmological Principle, based on their spatial distribution as a whole. We pointed out the large scale spatial pattern of the GRB activity does not necessarily re ects the large scale distribution of the cosmic matter.