Generalized Rescaled Pólya urn and its statistical applications

We introduce the Generalized Rescaled P\'olya (GRP) urn. In particular, the GRP urn provides three different generative models for a chi-squared test of goodness of fit for the long-term probabilities of correlated data, generated by means of a reinforcement mechanism. Beside this statistical application, we point out that the GRP urn is a simple variant of the standard Eggenberger-P\'olya urn, that, with suitable choices of the parameters, shows "local" reinforcement, almost sure convergence of the empirical mean to a deterministic limit and different asymptotic behaviours of the predictive mean.