{
  "abstract": "Statistical inference in sequential mediation models requires evaluating the cumulative distribution function (CDF) of the product of three normal coefficient estimators, an operation that entails integrating over a non-convex region with boundary $xyz=v$. While the Delta method provides an instantaneous first-order approximation, its asymptotic variance collapses whenever two or more path coefficients approach zero, producing severe undercoverage near parameter boundaries. Nonparametric bootstrapping avoids gradient collapse but incurs an $O(B \\times N)$ computational cost that becomes burdensome in large-scale simulation studies or iterative power analyses. We propose a model-based dimension-reduction algorithm that integrates out the third variable analytically under the trivariate Gaussian distribution for arbitrary mean vectors and positive-definite covariance matrices. Partitioning the $xy$-plane into quadrants isolates the sign change at the coordinate axes, reducing the problem to an adaptive two-dimensional quadrature whose fixed-grid cost is $O(n^2)$ in place of $O(n^3)$. In simulation benchmarks across six parameter regimes, the algorithm achieves a mean absolute error of $3.1 \\times 10^{-5}$ relative to a $10^{8}$-sample Monte Carlo reference, evaluating the distribution in under one second and yielding plug-in quantile confidence intervals whose empirical coverage was near or above nominal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05738",
  "author_total": 1,
  "authors": [
    {
      "name": "Davood Tofighi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7bnx8h9scav7p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05738",
  "fulltext_url": null,
  "id": "7bnx8h9scav7p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05738",
  "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",
      "stat.CO"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "openalex",
      "2613|Statistical Distribution Estimation and Applications"
    ]
  ],
  "openalex": "W7220665404",
  "pdf_url": "https://arxiv.org/pdf/2610.05738",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7bnx8h9scav7p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:16Z",
      "native_id": "2610.05738",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05738"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.CO",
      "subfield": "statistics"
    }
  ],
  "title": "Dimension Reduction for the Trivariate Normal Product Distribution: An Efficient Numerical Algorithm via Conditional Expectation",
  "title_text": "Dimension Reduction for the Trivariate Normal Product Distribution: An Efficient Numerical Algorithm via Conditional Expectation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/7bnx8h9scav7p",
  "venue": "arXiv",
  "versions": []
}