Parametrized Families of Gibbs Measures and their Statistical Inference

For Hölder continuous functions $f_i$, $i=0,\dots ,d$, on a subshift of finite type and $\mathrm{\Theta }\subset {\mathbb{R}}^d$ we consider a parametrized family of potentials $\{F_{\theta }=f_0+{\sum }_{i=1}^d{\theta }_i{f}_i:\theta \in \mathrm{\Theta }\}$. 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.