  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.
