Testing goodness-of-fit of random graph models

Random graphs are matrices with independent 0, 1 elements with probabilities determined by a small number of parameters. One of the oldest model is the Rasch model where the odds are ratios of positive numbers scaling the rows and columns. Later Persi Diaconis with his coworkers rediscovered the model for symmetric matrices and called the model beta. Here we give goodnes-of-fit tests for the model and extend the model to a version of the block model introduced by Holland, Laskey, and Leinhard

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 542 222 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 542 222 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 cannot 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 reflects the large-scale distribution of the cosmic matter