Motivated by recent studies of big samples, this work aims to construct a parametric model which is characterized by the following features: (i) a ‘local’ reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness-of-fit result for the limit probabilities. This triple purpose is achieved by the introduction of a new variant of the Eggenberger–Pólya urn, which we call the rescaled Pólya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited by the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.

The rescaled Pólya urn: local reinforcement and chi-squared goodness-of-fit test / G. Aletti, I. Crimaldi. - In: ADVANCES IN APPLIED PROBABILITY. - ISSN 0001-8678. - 54:3(2022 Sep 12), pp. 849-879. [10.1017/apr.2021.56]

The rescaled Pólya urn: local reinforcement and chi-squared goodness-of-fit test

G. Aletti
Primo
;
2022

Abstract

Motivated by recent studies of big samples, this work aims to construct a parametric model which is characterized by the following features: (i) a ‘local’ reinforcement, i.e. a reinforcement mechanism mainly based on the last observations, (ii) a random persistent fluctuation of the predictive mean, and (iii) a long-term almost sure convergence of the empirical mean to a deterministic limit, together with a chi-squared goodness-of-fit result for the limit probabilities. This triple purpose is achieved by the introduction of a new variant of the Eggenberger–Pólya urn, which we call the rescaled Pólya urn. We provide a complete asymptotic characterization of this model, pointing out that, for a certain choice of the parameters, it has properties different from the ones typically exhibited by the other urn models in the literature. Therefore, beyond the possible statistical application, this work could be interesting for those who are concerned with stochastic processes with reinforcement.
Empirical mean; central limit theorem; chi-squared test; compact Markov chain; Pólya urn; predictive mean; preferential attachment; reinforcement learning; reinforced stochastic process; urn model
Settore MAT/06 - Probabilita' e Statistica Matematica
Settore SECS-S/01 - Statistica
12-set-2022
Centro di Ricerca Interdisciplinare su Modellistica Matematica, Analisi Statistica e Simulazione Computazionale per la Innovazione Scientifica e Tecnologica ADAMSS
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/947251
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