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Exact harmonic mean p-value combination test

Mathematical Methods AIMech:ExactHarmonicMeanPValueTest DRAFT

Aliases: EHMP, Exact Harmonic Mean P-Value Test

A statistical p-value combination method that aggregates m hypothesis test p-values p_1, ..., p_m in (0, 1] with non-negative weights w_1, ..., w_m summing to 1 via the weighted reciprocal sum T_HMP(p) = sum_{j=1}^m w_j / p_j (whose terms X_j = 1 / p_j follow standard Pareto(1, 1) distributions on [1, infinity) under Uniform(0, 1)), evaluates the finite-sample null density and cumulative distribution function for arbitrary weights using Laplace-transform contour integration expressed via second-order exponential integrals (and switches to the unequal-weight Landau domain-of-attraction limit for large m), and inverts the test statistic to construct connected, strictly convex confidence regions.

Identity Notes & Concept Scope No normalized identity match for the Exact Harmonic Mean P-Value combination test across all 442 pinned AIO records and 566 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only frontier-tracking references in the Round 171 discovery report. Distinct from AIMech:HalfCauchyCombinationTest (HCCT), which maps p_j to [0, infinity) via tan((1 - p_j) * pi / 2) rather than to [1, infinity) via 1 / p_j and uses sine/cosine integral Laplace transforms rather than second-order exponential integral Laplace transforms. Distinct from the asymptotic equal-weight Harmonic Mean P-Value (HMP) approximation of Good (1958) and Wilson (2019), which applies the m -> infinity equal-weight Landau limit without exact finite-m unequal-weight Laplace contour integration or convex confidence-region inversion.

Claims & Grounded Evidence

4 Claims

Mechanism #pareto-reciprocal-statistic-and-laplace-contour-null

For individual p-values p_j in (0, 1] and non-negative weights w_1, ..., w_m summing to 1, EHMP forms T_HMP(p) = sum_{j=1}^m w_j / p_j, where each X_j = 1 / p_j has Pareto(1, 1) density f(x) = x^(-2) I_{x >= 1} under Uniform(0, 1), and computes the exact finite-m null density f_{Pareto, w}(x) and CDF for arbitrary unequal weights via Laplace-transform contour integration of X_j - 1 expressed in terms of the second-order exponential integral Ei_2(z) and Ramanujan's scaled series for Ei(x) / exp(x) to avoid numerical overflow.

supports Source: A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension · Locator: Section 2.1, lines 287–289; Section 2.3, lines 305–327; Appendix A, Theorem A.1, lines 1645–1654

Mechanism #unequal-weight-landau-domain-of-attraction-limit

For a triangular array of non-negative weights w_j^(m) summing to 1 with max_j w_j^(m) -> 0 as m -> infinity and i.i.d. Pareto(1, 1) variables X_j, the rescaled and shifted statistic (pi / 2) * (sum_{j=1}^m w_j^(m) X_j - A_m) converges in distribution to Landau(0, 1) with location shift A_m = 1 - gamma - sum_{j=1}^m w_j^(m) log w_j^(m), where gamma is the Euler-Mascheroni constant.

supports Source: A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension · Locator: Appendix A, Theorem A.2, lines 1656–1674; Appendix E, Proof of Theorem A.2, lines 2151–2160

Mechanism #convex-confidence-regions-via-pareto-convexity-dominance

Because the Pareto(1, 1) convexity transform dominates the standard Half-Cauchy transform on (0, 1), each reciprocal p-value term 1 / p_j(theta) for univariate t-tests (k_j >= 1) or multivariate Hotelling's T^2 tests (k_j >= d_j + 1) is strictly convex in theta, guaranteeing that inverting T_HMP(p(theta)) <= c_{m, 1 - alpha} yields a single connected finite interval in R and a bounded convex confidence region in R^d whenever the projection row spaces {Row(P_j)} span R^d.

supports Source: A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension · Locator: Section 3.1, lines 374–380; Appendix A, Theorem A.3, lines 1675–1680; Appendix E, Proof of Theorem A.3, lines 2161–2165

Application #large-p-insensitivity-and-tail-resilience-under-dependence

Because 1 / p_j >= 1 for all p_j in (0, 1] and approaches 1 as p_j -> 1, T_HMP(p) is bounded below by 1 and insensitive to large p-values near 1, while belonging to the alpha = 1 Landau domain of attraction so that extreme right-tail calibration remains resilient across dependent studies.

supports Source: A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension · Locator: Section 1.2, Table 1, lines 282–283; Section 2.4–2.5, Proposition 2.6, lines 341–368

⚠️ Curated Limitations & Negative Results

Direct evaluation of the second-order exponential integral Ei_2(z) in Theorem A.1 via standard special-function routines overflows for large real arguments because Ei(x) grows as exp(x) / x; numerical contour integration requires evaluating the exponentially scaled ratio Ei(x) / exp(x) (for example via Ramanujan's series expansion).
Because the individual Pareto(1, 1) terms X_j = 1 / p_j are bounded below by 1 with infinite mean, the null distribution of T_HMP(p) shifts upward at rate log m as the number of studies m grows, and applying the asymptotic equal-weight Landau approximation of Wilson (2019) at small m or under strongly unequal weights distorts finite-sample Type I error.
Like HCCT, inverting T_HMP(p(theta)) <= c_{m, 1 - alpha} produces an empty confidence set when inter-study inconsistency exceeds the critical threshold, and asymptotic tail validity under dependence is guaranteed in the extreme right tail (t -> infinity) rather than uniformly at moderate significance levels.
All 2,200 lines of Liu, Meng, & Pillai (2025, arXiv:2501.01065v1) were read. No author contour-integration package or empirical GWAS dataset was executed; constructed numerical checks in research/discovery/exact-harmonic-mean-pvalue-boundaries.py verify the exact m = 2 unequal-weight Pareto(1, 1) convolution density/CDF, large-p insensitivity, Example 1 confidence-interval convexity, and Theorem A.2 Landau location shifts only.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	The exact harmonic mean p-value combination test aggregates multiple (potentially dependent) hypothesis test p-values to control global Type I error and mitigate false-positive inflation under multiple testing (AIO:DataDredgingBias). AIO:DataDredgingBias denotes the multiple-testing interpretation bias rather than the statistical combination method.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`liu-meng-pillai2025`	A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension Kairong Liu, Xiao-Li Meng, Natesh S. Pillai	primary paper full_text	ID: https://doi.org/10.48550/arXiv.2501.01065 sha256:2fe912b58a74…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T09:58:00Z

Updated: 2026-10-17T09:58:00Z

Source File: data/curated/ExactHarmonicMeanPValueTest.yaml · Schema: curated_concept.yaml

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