{
  "abstract": "Equivariance is increasingly used in machine learning and statistics, often without systematic justification. In a companion article, the equivariance criterion was applied to the normal linear model with a fixed design matrix (fixed-$X$), yielding the minimum risk equivariant (MRE) estimators of the coefficient vector and of the condensed diagonal covariance matrix under a multivariate invariant location--scale group. We extend these results to the random-$X$ case, with covariates sampled from a population. The extension hinges on a distinction vacuous for fixed-$X$ but fundamental for random-$X$: whether risk and unbiasedness are evaluated conditionally on the realized design or after averaging over the design distribution. Under conditional evaluation, the fixed-$X$ group applies given $X$: least squares remains the best equivariant estimator of the coefficient vector, and the MRE estimators of the population variances keep their fixed-$X$ forms with population sizes at the realized design. Under absolute evaluation with an i.i.d.\\ design, the picture changes qualitatively: the natural scale group acting jointly on $(Y,X)$ fixes the coefficient vector, the induced parameter-space action is intransitive, equivariant risks are constant only along orbits indexed by the signal-to-noise ratio $ρ=\\|β\\|^2/σ^2$, and no uniformly minimum risk equivariant estimator exists. In the scalar case the optimal equivariant weight is the oracle shrinkage factor $w^*(ρ)=ρ/(ρ+E[T^{-1}])$, with least squares recovered as the infinite-signal limit $ρ\\to\\infty$---explaining and refining the known failure of the Gauss--Markov theorem with random regressors. For a centered design under location--scale transformations, least squares remains optimal within the natural invariant-contrast class, and the MRE estimator $S^2/(n-p+2)$ of the error variance is valid under both modes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16897",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhengyang Zhang"
    },
    {
      "name": "Haojin Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9m6w6yzbzz2zk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16897",
  "fulltext_url": null,
  "id": "9m6w6yzbzz2zk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16897",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Advanced Statistical Methods and Models"
    ]
  ],
  "openalex": "W7213408283",
  "pdf_url": "https://arxiv.org/pdf/2609.16897",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9m6w6yzbzz2zk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16897",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16897"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "The Equivariance Criterion in a Linear Model for Random-$X$ Cases",
  "title_text": "The Equivariance Criterion in a Linear Model for Random-X Cases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:51Z",
  "url": "https://research.draconitic.com/paper/9m6w6yzbzz2zk",
  "venue": "arXiv",
  "versions": []
}