{
  "abstract": "We study trend estimation in state-space models in which the trend has a fractional stochastic difference of order $d>0$ and the observation errors form a short-range-dependent stationary process. Using finite-sequence fractional summation and differencing operators, we analyze the penalized least-squares estimator obtained by shrinking the fractional differences of the trend. We derive asymptotic mean squared error (MSE) approximations for all $d>0$ and identify a sharp phase transition at $d=1/2$. When $d>1/2$, the estimator is consistent and its optimally balanced MSE has order $n^{-(2d-1)/(2d)}$. At the boundary $d=1/2$, we obtain a refined finite-sample approximation and show that the MSE decreases at the slower order $\\log\\log n/\\log n$. When $0<d<1/2$, the MSE converges to an explicit positive limit, so consistent recovery of the trend is impossible under the considered scaling. We also describe a practical criterion for choosing the penalty parameter and differencing order, and numerical experiments illustrate the MSE approximations and the behavior of the selection procedure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06727",
  "author_total": 3,
  "authors": [
    {
      "name": "Prabir Burman"
    },
    {
      "name": "Xiucai Ding"
    },
    {
      "name": "Robert H. Shumway"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9xcz592qjyq0x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06727",
  "fulltext_url": null,
  "id": "9xcz592qjyq0x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06727",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.CO"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Inference"
    ]
  ],
  "openalex": "W7212069113",
  "pdf_url": "https://arxiv.org/pdf/2609.06727",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9xcz592qjyq0x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06727",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06727"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Phase transitions and approximations of mean squared error for state-space models with fractional differencing",
  "title_text": "Phase transitions and approximations of mean squared error for state-space models with fractional differencing",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/9xcz592qjyq0x",
  "venue": "arXiv",
  "versions": []
}