How to estimate the memory of the Elephant Random Walk

TL;DR

This paper presents a new quasi-maximum likelihood estimator for the memory parameter of the elephant random walk, demonstrating its convergence and statistical properties across different regimes.

Contribution

It introduces a novel estimation method based on Taylor approximation and establishes its asymptotic properties for the elephant random walk.

Findings

Estimator converges almost surely in all regimes.

Asymptotic normality in the diffusive regime.

Asymptotic mixed normality in the superdiffusive regime.

Abstract

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.