Asymptotic normality and efficiency of the maximum likelihood estimator for the parameter of a ballistic random walk in a random environment

TL;DR

This paper analyzes the asymptotic properties of the maximum likelihood estimator for a parameter in a ballistic random walk within a random environment, demonstrating its normality, efficiency, and practical confidence region behavior.

Contribution

It proves the asymptotic normality and efficiency of the MLE for the parameter in a ballistic random walk in a random environment, with simulation validation.

Findings

MLE is asymptotically normal as the target site recedes.

The estimator achieves the Cramér-Rao lower bound.

Simulation confirms the practical effectiveness of confidence regions.

Abstract

We consider a one dimensional ballistic random walk evolving in a parametric independent and identically distributed random environment. We study the asymptotic properties of the maximum likelihood estimator of the parameter based on a single observation of the path till the time it reaches a distant site. We prove an asymptotic normality result for this consistent estimator as the distant site tends to infinity and establish that it achieves the Cram\'er-Rao bound. We also explore in a simulation setting the numerical behaviour of asymptotic confidence regions for the parameter value.