Open Access
2022 Generalized Rescaled Pólya urn and its statistical application
Giacomo Aletti, Irene Crimaldi
Electron. J. Statist. 16(1): 1635-1680 (2022). DOI: 10.1214/22-EJS1993


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.

Funding Statement

Partially supported by the Italian “Programma di Attività Integrata” (PAI), project “TOol for Fighting FakEs” (TOFFE) funded by IMT School for Advanced Studies Lucca.


We are grateful to the Referees for their useful comments.


Download Citation

Giacomo Aletti. Irene Crimaldi. "Generalized Rescaled Pólya urn and its statistical application." Electron. J. Statist. 16 (1) 1635 - 1680, 2022.


Received: 1 June 2021; Published: 2022
First available in Project Euclid: 8 March 2022

arXiv: 2010.06373
Digital Object Identifier: 10.1214/22-EJS1993

Primary: 60F05 , 60F15 , 62F03
Secondary: 62F05 , 62L20

Keywords: central limit theorem , Chi-squared test , Pólya urn , reinforced stochastic process , reinforcement learning , stochastic approximation , urn model

Vol.16 • No. 1 • 2022
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