Abstract
We introduce the Generalized Rescaled Polya (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-Polya 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.
Highlights
The standard Eggenberger-Polya urn has been widely studied and generalized
We note that the quantities p1, . . . , pk are related to the initial composition of the urn, but it can be seen as a long-term probability distribution on the possible values {1, . . . , k}
Under the proposed Generalized Rescaled Polya (GRP) urn model and the null hypothesis, the aggregate statistics L=1 Q has distribution Γ( L−2 1, 21λ ) and the corresponding p-value associated to the data is equal to 0.4579297
Summary
The standard Eggenberger-Polya urn (see [28, 40]) has been widely studied and generalized (for instance, some recent variants can be found in [6, 7, 14, 18, 21, 38, 39, 40]). The GRP urn provides a theoretical framework for a chi-squared test of goodness of fit for the long-term probabilities of correlated data, generated according to a reinforcement mechanism. Pk are related to the initial composition of the urn (since bi is the number of balls of color i that remains constant along time, while the quantity Bn i is updated according to the reinforcement mechanism), but it can be seen as a (typically unknown) long-term probability distribution on the possible values (colors) {1, . We note that the quantities p1, . . . , pk are related to the initial composition of the urn (since bi is the number of balls of color i that remains constant along time, while the quantity Bn i is updated according to the reinforcement mechanism), but it can be seen as a (typically unknown) long-term probability distribution on the possible values (colors) {1, . . . , k}
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
