BERCU, Bernard; LAULIN, Lucile

doi:10.1080/03610926.2022.2139149

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BERCU, Bernard
Institut de Mathématiques de Bordeaux [IMB]

LAULIN, Lucile
Institut de Mathématiques de Bordeaux [IMB]

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Communications in Statistics - Theory and Methods. 2022-11-01p. 1-21

Taylor & Francis

Résumé en anglais

We introduce an original way to estimate the memory parameter of the elephant random walk, a fascinating discrete time random walk on integers having a complete memory of its entire history. Our estimator is nothing more than a quasi-maximum likelihood estimator, based on a second order Taylor approximation of the log-likelihood function. We show the almost sure convergence of our estimate in the diffusive, critical and superdiffusive regimes. The local asymptotic normality of our statistical procedure is established in the diffusive regime, while the local asymptotic mixed normality is proven in the superdiffusive regime. Asymptotic and exact confidence intervals as well as statistical tests are also provided. All our analysis relies on asymptotic results for martingales and the quadratic variations associated.< Réduire

Mots clés en anglais

Elephant random walk

Maximum likelihood estimation

Almost sure convergence

Local asymptotic normality

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How to estimate the memory of the Elephant Random Walk

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