{
  "abstract": "A system contains an unknown number $N$ of units, each failing at an unknown common rate, each unit is removed once it fails. The failure-rate declines as units are removed, but after only a few failures the decline is too slight to distinguish from a constant rate, and the maximum-likelihood estimate of $N$ is infinite with substantial probability. This is a well-known feature of the model, but its probability has not been quantified. We show that about $N^{2/3}$ failures are needed before a finite population can be distinguished from an inexhaustible one, and that at this scale the profile log-likelihood ratio converges to that of a Gaussian shift experiment, in which an infinite estimate occurs with a calculable probability. The exponent is not a property of the model alone. Profiling annihilates the Taylor terms that the nuisance-parameters can reproduce, and detectability is governed by the first surviving term. For example, unknown drift in testing-effort reproduces the linear term, leaving a quadratic survivor and raising the requirement to $N^{4/5}$; in general the critical scale is $N^{2q/(2q+1)}$ for a survivor of order $q$. We obtain the limiting-law of the estimator, the null likelihood ratio distribution, and confidence sets that may be unbounded.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04453",
  "author_total": 1,
  "authors": [
    {
      "name": "Yakir Berchenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ftx574wkafyc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04453",
  "fulltext_url": null,
  "id": "4ftx574wkafyc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04453",
  "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.AP"
    ],
    [
      "arxiv",
      "stat.OT"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04453",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ftx574wkafyc.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04453",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04453"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Nuisance structure and the detection boundary for population size: how profiling can change a rate, not only a constant",
  "title_text": "Nuisance structure and the detection boundary for population size: how profiling can change a rate, not only a constant",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:10Z",
  "url": "https://research.draconitic.com/paper/4ftx574wkafyc",
  "venue": "arXiv",
  "versions": []
}