Parametrized Families of Gibbs Measures and their Statistical Inference

TL;DR

This paper studies parametrized Gibbs measures on subshifts, proving the consistency and asymptotic normality of maximum likelihood estimators, and explores applications in statistical inference and connections to Markov chains.

Contribution

It establishes the statistical properties of MLE for Gibbs measures with parametrized potentials on subshifts, including consistency and asymptotic distribution.

Findings

MLE of parameters is consistent.

Asymptotic normality of the estimator is proven.

Applications include confidence intervals and hypothesis testing.

Abstract

For H\"older continuous functions $f_i$, $i=0,\ldots ,d$, on a subshift of finite type and $\Theta\subset \mathbb \R^d$ we consider a parametrized family of potentials $\{F_\theta= f_0+\sum_{i=1}^d \theta_i f_i : \theta\in \Theta\}$. We show that the maximum likelihood estimator of $\theta$ for a family of Gibbs measures with potentials $F_\theta$ is consistent and determine its asymptotic distribution under the associated shift-invariant distribution. A second part discusses applications; from confidence intervals through testing problems to connections to Bernoulli distributions and stationary Markov chains.