{
  "id": "2506.21527",
  "version": "v1",
  "published": "2025-06-26T17:48:18.000Z",
  "updated": "2025-06-26T17:48:18.000Z",
  "title": "Asymptotic Inference for Exchangeable Gibbs Partition",
  "authors": [
    "Takuya Koriyama"
  ],
  "comment": "35 pages, 3 figures",
  "categories": [
    "math.ST",
    "math.PR",
    "stat.TH"
  ],
  "abstract": "We study the asymptotic properties of parameter estimation and predictive inference under the exchangeable Gibbs partition, characterized by a discount parameter $\\alpha\\in(0,1)$ and a triangular array $v_{n,k}$ satisfying a backward recursion. Assuming that $v_{n,k}$ admits a mixture representation over the Ewens--Pitman family $(\\alpha, \\theta)$, with $\\theta$ integrated by an unknown mixing distribution, we show that the (quasi) maximum likelihood estimator $\\hat\\alpha_n$ (QMLE) for $\\alpha$ is asymptotically mixed normal. This generalizes earlier results for the Ewens--Pitman model to a more general class. We further study the predictive task of estimating the probability simplex $\\mathsf{p}_n$, which governs the allocation of the $(n+1)$-th item, conditional on the current partition of $[n]$. Based on the asymptotics of the QMLE $\\hat{\\alpha}_n$, we construct an estimator $\\hat{\\mathsf{p}}_n$ and derive the limit distributions of the $f$-divergence $\\mathsf{D}_f(\\hat{\\mathsf{p}}_n||\\mathsf{p}_n)$ for general convex functions $f$, including explicit results for the TV distance and KL divergence. These results lead to asymptotically valid confidence intervals for both parameter estimation and prediction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-26T17:48:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exchangeable gibbs partition",
      "asymptotic inference",
      "parameter estimation",
      "maximum likelihood estimator",
      "generalizes earlier results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}