The trace-gamma distribution: a three-cumulant calibrated reference for quadratic-form tests, with Welch's F and Pearson's χ² as special cases (mdist)— reproducibility deposit

Abstract

The trace-gamma distribution: a three-cumulant calibrated reference for quadratic-form tests, with Welch's F and Pearson's χ² as special cases William J. Dwyer, MD, MPH, FAAP — Department of Mathematics and Statistics, University of Massachusetts Lowell. ORCID 0009-0004-0855-7222. Concept DOI (always resolves to the latest version): 10.5281/zenodo.22060095. Published v1.0.13:10.5281/zenodo.22143061. What this is The reproducibility deposit for the trace-gamma paper. Many everyday tests are quadratic forms in approximately-normal estimates — Welch's heteroscedastic F, Pearson's χ² of independence, score and Wald statistics — and their exact null is a generalized chi-square (a weighted mix of χ²'s) that software almost always replaces with a one- or two-moment approximation. Those approximations miscalibrate exactly where it matters: few groups, unequal variances, sparse or skewed data. The paper introduces the trace-gamma distribution, a three-cumulant calibrated reference built from the quadratic form's own mean, variance, and third cumulant via the Lancaster cumulant-trace identity, and shows that Welch's F and Pearson's χ² fall out as special cases. It matches the exact Imhof (1961) inversion across a wide design grid, carries an estimable non-Gaussian cumulant correction, and — deployed as a test reference — is a deflation of the naive two-moment call (its rejection region is a strict subset, so a significant result can only be withdrawn, never manufactured; no reverse flip is possible). What the deposit contains Manuscript (author + anonymized; built .docx/.pdf), the novelty / prior-art companion, and the long-form derivations companion (D1–D8, proof-complete: the generalized-chi-square null, the Lancaster cumulant-trace identity, the trace-gamma parameterization, and the arguments behind Propositions 1–6). Reproducibility apparatus — the distribution object mseries_qgamma.py (TraceGamma, constructors from cumulants), the four cross-checks validate_qgamma.py, the base experiments qgamma_experiments.py (V1–V3), the estimable cumulant correction, the exact Imhof-inversion cross-check qgamma_imhof.py, the wider design grid (group size × imbalance × input shape), and the deployed stress grid — each with its locked CSV. Every reported number regenerates from these deterministically-seeded scripts. Figures — the accuracy panels, the cumulant-correction figure, the design-grid and stress-grid figures, and the Imhof cross-check. Interactive demonstrator honest_tracegamma.html — computes the two-cumulant-vs-trace-gamma verdict flip and its population incidence in the browser, reproduces the deposited Python, and carries the house flip-interpretation standard (deflation marker, "how to read a flip" beat, incidence panel, assertNoReverse). Deep-dive record and the submission apparatus. All evaluation is simulation-based with an exact-inversion cross-check. Code is released under the MIT License; text, figures, and data under CC BY 4.0. How to cite Please cite this deposit if you use the package or the method. Citing the concept DOI references the work in general and always resolves to the latest version; cite a specific version DOI to point at an exact snapshot. Dwyer, W. J. (2026). The trace-gamma distribution: a three-cumulant calibrated reference for quadratic-form tests — reproducibility deposit [Software]. Zenodo. https://doi.org/10.5281/zenodo.22060095 BibTeX: bibtex @software{dwyer_tracegamma_2026, author = {Dwyer, William J.}, title = {The trace-gamma distribution: a three-cumulant calibrated reference for quadratic-form tests --- reproducibility deposit}, year = {2026}, publisher = {Zenodo}, doi = {10.5281/zenodo.22060095}, url = {https://doi.org/10.5281/zenodo.22060095}, orcid = {0009-0004-0855-7222} } The DOI above is the concept DOI (resolves to the latest version); to cite a specific release use that version's DOI in place of it (e.g. 10.5281/zenodo.22143061 for v1.0.13). When the accompanying journal article appears, please cite it as the primary reference for the method and this deposit as the reproducibility archive. Version history v1.0.14 — ✅ 10.5281/zenodo.22187379 (2026-08-31) staged (pending upload): rendering-only refresh — the manuscript and Derivations Companion docx/PDF rebuilt through the current math-typography builder so nested-paren radicals draw as true Office-math radicals; the deposit now also carries the current shared house tooling (the bundled mseries_deposit.py includes the require_all deposit guard). No number, figure, table, or claim changed; the 71 in-text cross-reference links are intact. New version on concept 10.5281/zenodo.22060095 (tracegamma_reproducibility_v1.0.14.zip, md5 6f6a0e5c1e29d6cd3fa34008458666f0, 9,937,855 B, 59 files). v1.0.13 — ✅ 10.5281/zenodo.22143061 (2026-08-28). v1.0.12 — ✅ 10.5281/zenodo.22142142 (2026-08-28): two companion documents added — the novelty/prior-art review and the full-detail D1–D8 derivations ladder; §6 now points to the derivations companion; body unchanged. v1.0.11 — ✅ 10.5281/zenodo.22132594 (2026-08-27): cover letter brought to the house standard; first fully deterministic (byte-reproducible) deposit. v1.0.10 — ✅ 10.5281/zenodo.22125099 (2026-08-27): restored the in-text cross-reference hyperlinks in the Word/PDF. v1.0.0 — first deposit (10.5281/zenodo.22060096; concept 10.5281/zenodo.22060095): the trace-gamma distribution object, the Imhof cross-check, the design grid, and the honest_tracegamma.html demonstrator. Provenance: every number traces to a named, deterministically-seeded script and is cross-checked against the exact Imhof (1961) inversion; the demonstrator reproduces the deposited Python. Related records: the T_root methodology 10.5281/zenodo.21522471; the ANOVA sibling m01A 10.5281/zenodo.21908169.