Generalized Rescaled Pólya urn and its statistical application

Abstract

We introduce the Generalized Rescaled Pólya (GRP) urn, that provides a generative model for a chi-squared test of goodness of fit for the long-term probabilities of clustered data, with independence between clusters and correlation, due to a reinforcement mechanism, inside each cluster. We apply the proposed test to a data set of Twitter posts about COVID-19 pandemic: in a few words, for a classical chi-squared test the data result strongly significant for the rejection of the null hypothesis (the daily long-run sentiment rate remains constant), but, taking into account the correlation among data, the introduced test leads to a different conclusion. Beside the statistical application, we point out that the GRP urn is a simple variant of the standard Eggenberger-Pólya 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. Moreover, the study of this model provides the opportunity to analyze stochastic approximation dynamics, that are unusual in the related literature.

Incidence rate estimation of SARS-COVID-19 via a Polya process scheme: a comparative analysis in Italy and European countries

“…Unfortunately, this is not empirically possible, since different trials of the same i-th country (region) are in principle inaccessible. In a recent paper [21] the authors proposed a rescaled version of Polya model, that, taking into account the correlation among data, due to the reinforcement mechanism, shows almost sure convergence of the empirical mean, under suitable conditions. However, since we are interested in deriving the shape of the distribution of the confirmed cases and of the fraction 𝜌(𝑛), we adopt a different strategy.…”

The Rescaled Polya Urn and the Wright-Fisher process with mutation

“…The parameter α regulates the reinforcement mechanism: the greater α, the greater the dependence of Nn i on n h=1 ξ h i . In [1,2,3] the Rescaled Pólya (RP) urn has been introduced, studied, generalized and applied. This model is characterized by the introduction of a parameter β in the original model so that Nn i = bi + Bn i with…”

“…This aspect is important for the statistical applications in the context of a "big sample", when a small value of the chi-squared distance might be significant, and hence a correction related to the correlation between observations is desirable. In [1,2] it is described a possible application in the context of clustered data, with independence between clusters and correlation, due to a reinforcement mechanism, inside each cluster.…”