Consistency of maximum likelihood estimation for some dynamical systems

TL;DR

This paper proves the asymptotic consistency of maximum likelihood estimation for certain noisy dynamical systems, linking dynamical properties with statistical estimation theory.

Contribution

It establishes conditions under which maximum likelihood estimation is consistent for dynamical systems, connecting dynamical properties with statistical inference.

Findings

MLE is consistent for shifts of finite type with Gibbs measures

MLE is consistent for Axiom A attractors with SRB measures

Provides a unified framework linking dynamical systems and statistical estimation

Abstract

We consider the asymptotic consistency of maximum likelihood parameter estimation for dynamical systems observed with noise. Under suitable conditions on the dynamical systems and the observations, we show that maximum likelihood parameter estimation is consistent. Our proof involves ideas from both information theory and dynamical systems. Furthermore, we show how some well-studied properties of dynamical systems imply the general statistical properties related to maximum likelihood estimation. Finally, we exhibit classical families of dynamical systems for which maximum likelihood estimation is consistent. Examples include shifts of finite type with Gibbs measures and Axiom A attractors with SRB measures.