Skip to content
Log in
#edge computing Open access

The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA. (T_BB, m01a): Methods, confidence-set-geometry development, derivations, novelty, and reporting standard): reproducibility deposit

William Dwyer
Aug 2026 · Zenodo (CERN European Organization for Nuclear Research) · 3 references

Abstract

The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA 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.21908169. What this is The reproducibility deposit for the deflated-Welch statistic T_BB, a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA (the Behrens–Fisher problem for k ≥ 3 groups). Welch's test becomes liberal under skew and unstable variance weights at small samples; T_BB = Q(s²)·exp(−R) keeps the ordinary group means and buys a guaranteed level by deflating the Welch quadratic by a Berger–Boos scale-inflation radius R. Three operating points are provided: a fixedcalibrated radius (κ_s), a design-adaptive near-guarantee radius (closed-form polygamma Cornish–Fisher with a finite-nkurtosis guard), and a fully proved smallest-eigenvalue radius R_eig (Gaussian, extended under bounded kurtosis). What the deposit contains Manuscript (author + anonymized) and a derivations supplement (DA1–DA13) plus a long-form derivations companion, covering: why Welch fails under skew in closed form; the Berger–Boos deflation and its exact worst-case radius; the polygamma-cumulant Cornish–Fisher radius with saddlepoint-exact normal backbone; the excess-kurtosis tail term with its finite-n upper-confidence guard; the imbalance correction; the fully proved smallest-eigenvalue radius (with the k-group multiplicity fix, free-β optimization, and the proved-under-bounded-kurtosis widening); and the k-sample Behrens–Fisher null distribution. Interactive demonstrator rerun_cochran/honest_anova.html — computes raw-mean Welch, the fixed / adaptive / proved T_BB radii, the estimand-changing transform routes, and the full routing receipt in the browser, reproducing the deposited Python. Its engine is extracted as a standalone Node module (m01A_anova_engine.js) and checked cell-by-cell against Python across an 84-design taxonomy (verify_anova_engine_taxonomy.py/.js, max |Δp| = 0.00000). Reproducibility scripts (rerun_cochran/, rerun/) — every reported number traces to a named, deterministically-seeded script (size/power/surface, the calibration and information-limit decompositions, the proved-radius verification, the imbalance calibration, the skew-router branch, and the figures). Real-data evidence — anova_flip_scan.py scans 2,783 public one-way layouts (254 datasets): guaranteed T_BBwithholds ~41% of Welch-significant calls, concentrated where the weight-instability screen fires, and never manufactures significance (Table 7 / Figure 15). Figures and the deterministic deposit builder (fixed timestamps → stable md5). All evaluation is simulation-based; the one empirical component is the public-dataset scan, which uses only openly distributed data. Code is released under the MIT License; text and figures under CC BY 4.0. Version history (consolidated changelog) Published version DOIs are marked ✅; the concept DOI above always resolves to the latest. Staged versions were rolled into the next published one unless noted. v1.0.74 (2026-08-29) — Proved-under-bounded-kurtosis radius (DA12.6). The proved non-normal widening now keys on excess kurtosis, √(1 + κ̂·(n−1)/(2n)), from the exact Var(s²/σ²) = 2/(n−1) + κ/n, so symmetric heavy tails (Student-t) are covered where the old skew form √(1 + 0.75·skew²) under-covered; tbbProved switched to the kurtosis form across the demonstrator, engine, and Python truth (re-verified JS-vs-Python at 0.00000); new rc_anova_kurtosis_proof.py + deep-dive. v1.0.73 ✅ 10.5281/zenodo.22165709 (2026-08-29) — Reconstructed & verified demonstrator engine (standalone Node module + taxonomy verifier, max |Δp| = 0.00000 across 84 designs; Yuen zero-variance fix; T_BB-routed presets both directions); series-impact deep-dive (the corrected R_eig k-group multiplicity gap also reaches m03 and m01t). v1.0.72 (2026-08-29) — Title set to "The deflated-Welch statistic…"; corrected + optimized proved radius R_eig (β/k multiplicity fix + β-optimization, DA12); real-data Welch-vs-T_BB flip scan (2,783 layouts; Table 7 / Figure 15) + demonstrator imbalance-factor fix; long-form derivations companion. v1.0.71 / v1.0.70 (2026-08-21) — Zhang normal-reference comparator benchmarked on the efficiency frontier (valid on only 24% of designs, in the calibrated-liberal cluster); k = 2 adaptive-radius case-study fold (design-scaling vs shape-keying distinction). v1.0.69 ✅ 10.5281/zenodo.22035826 (2026-08-20) — HTML R1/R2 presentation pass + Figure 9 adaptive per-cluster label merge. v1.0.68 ✅ 10.5281/zenodo.22033737 (2026-08-20) — Companion consolidation into a single six-column Table 6; Figures 11–14 harmonized into one story. v1.0.67 / v1.0.65 / v1.0.60 (2026-08-19/20) — Guarded-reference naming-collision fix; the 40,000-replication expanded-frontier pin (Table 3 + Figure 8) with the symmetric-heteroscedastic skew-router branch; the mean-preserving lightened-R_eig do-not-use fallback. v1.0.59 ✅ 10.5281/zenodo.21995320 (2026-08-18) — Reporting standard + honest_anova.html demonstrator re-aligned to the current T_BB methods paper. v1.0.57 ✅ 10.5281/zenodo.21986847 (2026-08-17) — Reviewer-comprehension pass (multi-paragraph abstract, contributions list, trimmed captions); proved radius R_eig added as a Table 3 scorecard row; corner tail-index correction (N−k)/2 (low-order moments exist in every deployed design). v1.0.56–v1.0.49 (2026-08-16) — The k-sample Behrens–Fisher corner-distribution program: two-moment scaled-χ² corner reference, derived corner cumulants, the secular-eigenvalue law + closed CGF + power-law tail, consolidated into derivations DA13 with a prior-art/novelty audit. v1.0.48 ✅ 10.5281/zenodo.21963458 (2026-08-16) — The unifying λ(z) correction (a smooth instability-keyed deflation strength). v1.0.45 ✅ 10.5281/zenodo.21962965 (2026-08-16) — Atomic sparsity index + bootstrap-t edge hardening + shape-aware pooled standardized-residual bootstrap (SA-PSRB); multivariate transfer to m03. v1.0.44–v1.0.41 (2026-08-16) — Shape-moment re-injection order (skew is the sweet spot), validated and hardened pooled standardized-residual bootstrap, atomic weight-noise probes. v1.0.40 ✅ 10.5281/zenodo.21961667 (2026-08-16) — Log-domain weight-stabilization probe (negative for stabilization; clarifies the size-adjusted oracle ceiling); includes the oracle-power gap decomposition (≈92% conservatism, ≈8% estimation). v1.0.37 ✅ 10.5281/zenodo.21961327 (2026-08-16) — Residual-bootstrap qualification of the shoot-out + the first proved Gaussian smallest-eigenvalue radius R_eig (DA12, the p = 1 specialization of the m03 theorem). v1.0.36 (2026-08-15) — Figure 11 T_BB-region colour fix (amber, matching the routing figures). v1.0.27 ✅ 10.5281/zenodo.21908170 — Earlier published baseline of the deposit. Provenance: every number traces to a named, deterministically-seeded script listed in the manuscript Declarations; the demonstrator engine reproduces the deposited Python to max |Δp| = 0.00000 across the taxonomy verification. License. Code and scripts in the deposit are released under the MIT License; text and figures under CC BY 4.0. Reuse is permitted with attribution to the author and citation of the concept DOI above. How to cite. Dwyer, W. J. The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA. Reproducibility deposit, Zenodo. https://doi.org/10.5281/zenodo.21908169

DOI
View source

Similar papers

#federated learning Sep 2026

AoI Minimization in Heterogeneous MEC Networks: A Federated Learning-Assisted Hybrid DRL and Convex Approach

This paper investigates a dynamic heterogeneous mobile edge computing network (HMECN), where mobile devices (MDs) could offload their full tasks to a small base station (SBS) directly or the macro base station (MBS) in direct or relay mode. As age of information (AoI) is a comprehensive and accurate metric to capture the freshness of computation results, we formulate a long-term weighted sum AoI (LWSA) minimization problem in the HMECN by jointly optimizing the offloading decisions of MDs as well as the bandwidth and computation resource allocation of all base stations, subject to energy, delay and peak AoI constraints. To address the formulated non-convex mixed integer nonlinear programming problem, we decompose it into the offloading decision optimization (ODO) top-problem and the resource allocation optimization (RAO) sub-problem. Based on the decomposition, we propose a federated learning (FL)-assisted hybrid DRL and convex approach that is comprised of a safe multi-agent DRL algorithm, convex optimization and FL. The ODO top-problem is solved by the safe multi-agent DRL algorithm, which strictly ensures that the actions of each agent do not exceed its energy constraint and then paves the way for using convex optimization to solve the RAO sub-problem. FL is used to alleviate the training instability problem aggravated by multi-agent settings via breaking the limitation of partial knowledge for each individual agent. Simulation results demonstrate the superiority of the proposed approach in terms of the LWSA, convergence, scalability and robustness in dynamic environments.

Xiaoying Liu, Junhao Zheng, Kechen Zheng et al. · 8 citations
#edge computing Sep 2026

DRUDM-CFG: A Fairness-Aware Multi-Agent DRL Algorithm for AMEC-Assisted Task Offloading in Post-Disaster Scenarios

High-altitude airships (HAS) and uncrewed aerial vehicles (UAVs) equipped with Multiaccess Edge Computing (MEC) servers have emerged as promising aerial MEC nodes for providing task offloading (TO) services to intelligent mobile devices (IMDs) in post-disaster scenarios. HAS offers robust computing and energy resources, while UAVs provide flexible, low-altitude coverage for rapid deployment. However, direct task offloading from IMDs to HAS often leads to task failures due to high transmission delays. UAVs with limited onboard resources require to minimize resource waste. Additionally, IMDs in sparse areas face insufficient TO services due to unfair UAV coverage. This paper defines these challenges as a joint optimization problem involving TO, RA, and UAV coverage fairness. It proposes a cooperative aerial Multiaccess Edge Computing (AMEC) framework integrating HAS and UAVs to address the issue. Within this framework, a hybrid TO scheme is first developed to mitigate the high transmission delay between IMDs and HAS. Second, a Distance, Resource, Urgency-based Decision Mechanism (DRUDM) is designed to enhance the accuracy of UAVs in selecting target IMDs for TO services. Third, a Coverage Fairness Guarantee (CFG) strategy is proposed to optimize UAV flight trajectories, ensuring IMDs in sparse areas receive fair TO services. Finally, the joint optimization problem is modeled as a Multi-Agent Partially Observable Markov Decision Process (MA-POMDP), and a DRUDM–CFG algorithm is presented to efficiently solve this complex non-convex optimization problem. Experimental results demonstrate that the proposed algorithm outperforms other compared algorithms in task completion rate and average delay, benefiting from the DRUDM mechanism. Meanwhile, the CFG strategy effectively improves TO service fairness for IMDs in sparse areas.

Xiting Peng, Chuanqi Qin, Xiaoyu Zhang et al. · 4 citations
#edge computing Preprint Aug 2026

Computing an e-net of a closed hyperbolic surface

Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips. The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part. As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.

V. Delecroix, Vincent Despré, Camille Lanuel et al. · 3 citations
#edge computing Sep 2026

Toward 6G Edge Intelligence: Lightweight LLMs for Intent-Driven Network Automation

Future 6G networks are envisaged to tightly integrate communication, sensing, and computing, demanding real-time, intent-driven intelligence at the edge. While large language models (LLMs) excel in intent recognition and semantic reasoning, their application to real-time network lifecycle management at the edge is limited by heterogeneous application intents (APPIs), dynamic network conditions, and severe resource constraints. This paper proposes a novel lightweight LLM architecture, KGLlama-KD, that synergizes knowledge graphs (KGs) with knowledge distillation (KD) to enable intent-driven networking and enhance 6G edge intelligence. Specifically, a KG is constructed to formally describe the relationships among application scenarios, functional primitives, performance requirements within APPIs, and the correspondences between APPIs and network service requests (NSRs), thereby producing a structured intent training dataset. Building upon the Llama 3 foundation model, a two-phase optimization framework is designed to support lightweight edge deployment while preserving translation fidelity. The LLM is first fine-tuned with KG guidance and compressed via KD in the cloud, and then deployed on resource-constrained edge nodes to perform real-time, accurate, and efficient APPIs interpretation. Experiments validate that KGLlama-KD achieves 95% accuracy for APPI understanding, surpassing DeepSeek and Qwen by an average of 8%. The distilled model reduces inference latency by 60% compared to full-scale LLMs, fulfilling the sub-100 ms requirement for 6G latency-sensitive services.

Bing Wu, Sai Zou, Minghui Liwang et al. · 3 citations
#edge computing Sep 2026

Joint Latency and Charge Cost Minimization for Reliable Task Offloading in Dispersed Computing: A Multi-Objective Optimization Approach

Dispersed computing has emerged as a promising paradigm that leverages underutilized resources from massive Internet of Things devices (IoTDs) to enhance the computing capacity at the network edge. However, existing works about the dispersed computing overlook the heterogeneous computing environment with parallel and serial computations and task reliability requirements for the hardware-constrained IoTDs, and they lack multi-objective optimization approaches to optimize the task offloading. To address the challenges, we propose a comprehensive scheme to achieve a delay-aware and economic-aware dispersed computing paradigm by using a multi-objective optimization approach. Particularly, we consider parallel processing at an edge server and serial processing at the lightweight IoTDs, and leverage the task redundancy to satisfy the task reliability requirements on the IoTD side. We further formulate a constrained multi-objective optimization problem (CMOP) aiming at jointly optimizing the task assignment, bandwidth allocation, and CPU frequency allocation to simultaneously minimize the total delay cost and the total charge cost of the tasks. To address the CMOP, we propose an improved constrained multi-objective evolutionary algorithm that employs a dual-population cooperative mechanism between two populations and a repairing constraint-handling technique. The dual-population cooperative mechanism can balance convergence toward Pareto optimality and solution diversity maintenance. The repairing constraint-handling technique is designed to guide solutions toward feasible regions, achieving efficient exploration of complex constrained search spaces. Simulation results demonstrate the superiority of our algorithm in seeking the better-converged and better-distributed Pareto optimal solutions to well address the tradeoffs between the two objectives.

Xumin Huang, Zexiong Wu, Chaoda Peng et al. · 2 citations
#edge computing Sep 2026

Service Enhancement and Reliability Assurance in 6G Vehicular Networks via a Stackelberg Game-Theoretic Approach

With the rapid development of 6G and Internet of Vehicles (IoV) technologies, the volume of computation-intensive tasks generated by intelligent vehicles is growing exponentially. Given limited onboard processing capabilities, vehicles increasingly rely on edge servers deployed by service providers (SPs) at roadside units to offload tasks. Vehicle clients can offload the tasks to SPs to mitigate their onboard computation load, while SPs derive economic benefits through the provision of computation resources. However, this interaction introduces a conflict of interest, as vehicles aim to minimize their offloading costs, while SPs seek to maximize revenue. To address this problem, we propose SPOR, a Stackelberg game-based service priority-aware computation offloading and resource pricing scheme in IoV. SPOR is a hierarchical game-theoretic framework in which SPs act as leaders setting prices, while vehicles act as followers determining their offloading strategies. A novel service prioritization function is introduced, incorporating booking price, system load, and reputation to ensure fair and balanced resource allocation. We provide a theoretical proof of the existence and uniqueness of a Nash equilibrium. Extensive experiments on a real-world vehicle edge computing dataset show that SPOR outperforms baseline methods in delay, energy consumption, average load, and task completion rate. Notably, SPOR maintains task completion rates above 97% even under heavy workloads, demonstrating its effectiveness in enhancing system reliability and overall performance.

Kai Peng, Yuanlin Lin, Shuai Zhao et al. · 2 citations

Related blog posts