How to estimate the memory of the Elephant Random Walk

Bernard Bercu (1) , Lucile Laulin (1)

1 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: Elephant random walk; Maximum likelihood estimation; Almost sure convergence; Local asymptotic normality

Domaines: Statistiques [math.ST]; Probabilités [math.PR]; Analyse de données, Statistiques et Probabilités [physics.data-an]

Bernard Bercu, Lucile Laulin. How to estimate the memory of the Elephant Random Walk. Communications in Statistics - Theory and Methods, 2022, pp.1-21. ⟨10.1080/03610926.2022.2139149⟩. ⟨hal-03483257⟩