Parametrized Families of Gibbs Measures and their Statistical Inference

For H\"older continuous functions fᵢ, i=0,… ,d, on a subshift of finite type and Θ⊂ ᵈ we consider a parametrized family of potentials _θ= f₀+∑ᵢ₌₁ᵈ θᵢ fᵢ : θ∈ Θ\. We show that the maximum likelihood estimator of θ for a family of Gibbs measures with potentials F_θ 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.