{
  "id": "2408.01104",
  "version": "v1",
  "published": "2024-08-02T08:28:54.000Z",
  "updated": "2024-08-02T08:28:54.000Z",
  "title": "Parametrized Families of Gibbs Measures and their Statistical Inference",
  "authors": [
    "Manfred Denker",
    "Marc Keßeböhmer",
    "Artur O. Lopes",
    "Silvia R. C. Lopes"
  ],
  "comment": "37 pages",
  "categories": [
    "math.DS",
    "math.ST",
    "stat.TH"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-02T08:28:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62F02",
      "62F12",
      "62E20",
      "37A50",
      "37A10"
    ],
    "keywords": [
      "gibbs measures",
      "statistical inference",
      "parametrized family",
      "second part discusses applications",
      "stationary markov chains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}