Cancellation-free distributional symmetry testing via reliability-inspired consecutive-system entropy scores

Ibrahim A. Nafisah; Tahani Alshathri; Mashael A. Alshehri; Ibrahim A. Nafisah; Tahani Alshathri; Mashael A. Alshehri

doi:10.3934/math.20261406

AIMS Mathematics

2026, Volume 11, Issue 11: 36293-36355. doi: 10.3934/math.20261406

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Cancellation-free distributional symmetry testing via reliability-inspired consecutive-system entropy scores

1.
Department of Statistics and Operations Research, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Department of Quantitative Analysis, College of Business Administration, King Saud University, Riyadh 11362, Saudi Arabia

Received: 12 September 2026 Revised: 12 October 2026 Accepted: 15 October 2026 Published: 04 November 2026
MSC : 62B10, 62G09, 62G10

Testing distributional symmetry about an unknown center is a composite-null problem in which signed information contrasts can cancel under asymmetric alternatives. We developed a two-layer framework built on reflected probability kernels from paired consecutive reliability systems: Global fractional-entropy identities characterize symmetry, while a positive-weight complementary log-quantile-density discrepancy measures local departures without cancellation. The discrepancy identified symmetry for any strictly positive weight; the consecutive-system kernels supplied a structured, reliability-interpretable choice rather than a necessary or optimal one. We proved consistency of the moving-spacing estimator under explicit conditions and gave a compact-support sufficient class with uniform control of all complete windows, including boundary windows. For finite-sample inference, we introduced ACES-R, a distinct blockwise reflection-randomization core with finite-sample level control over the composite symmetry null, without estimating the center. Sign-paired ties generally made its complete-orbit levels conservative, and fixed-block consistency held on the stated compact-support class. A separate Monte Carlo member used a prespecified quadratic statistic. Blocking carried an information cost: ACES-R exhibited lower raw rejection rates than full-sample ACES in the reported asymmetric settings, and its finite-group guarantee did not validate the full-sample bootstrap calibration. Full-sample ACES combined the score orders 1, 3/2, and 2 in a symmetrized-bootstrap standardized maximum whose primitive bootstrap validity and joint null limit remained unproved. Paired-sample simulations quantified the blocking-power trade-off and examined Student-t, Cauchy, and skew-t settings without claiming equal-size or uniform power superiority; classical and characteristic-function benchmarks, ablations, rounding diagnostics, and two engineering datasets mapped empirical strengths and limitations, with any heavy-tail power advantage remaining exploratory. Archived code, seeds, and replicate-level outputs support reproducibility of the paired-sample experiments.
- distributional symmetry,
- nonparametric symmetry testing,
- quantile density,
- consecutive-system weights,
- reliability systems,
- moving spacings,
- symmetrized bootstrap,
- randomization tests,
- finite-sample level control
Citation: Ibrahim A. Nafisah, Tahani Alshathri, Mashael A. Alshehri. Cancellation-free distributional symmetry testing via reliability-inspired consecutive-system entropy scores[J]. AIMS Mathematics, 2026, 11(11): 36293-36355. doi: 10.3934/math.20261406

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Abstract

Testing distributional symmetry about an unknown center is a composite-null problem in which signed information contrasts can cancel under asymmetric alternatives. We developed a two-layer framework built on reflected probability kernels from paired consecutive reliability systems: Global fractional-entropy identities characterize symmetry, while a positive-weight complementary log-quantile-density discrepancy measures local departures without cancellation. The discrepancy identified symmetry for any strictly positive weight; the consecutive-system kernels supplied a structured, reliability-interpretable choice rather than a necessary or optimal one. We proved consistency of the moving-spacing estimator under explicit conditions and gave a compact-support sufficient class with uniform control of all complete windows, including boundary windows. For finite-sample inference, we introduced ACES-R, a distinct blockwise reflection-randomization core with finite-sample level control over the composite symmetry null, without estimating the center. Sign-paired ties generally made its complete-orbit levels conservative, and fixed-block consistency held on the stated compact-support class. A separate Monte Carlo member used a prespecified quadratic statistic. Blocking carried an information cost: ACES-R exhibited lower raw rejection rates than full-sample ACES in the reported asymmetric settings, and its finite-group guarantee did not validate the full-sample bootstrap calibration. Full-sample ACES combined the score orders 1, 3/2, and 2 in a symmetrized-bootstrap standardized maximum whose primitive bootstrap validity and joint null limit remained unproved. Paired-sample simulations quantified the blocking-power trade-off and examined Student-t, Cauchy, and skew-t settings without claiming equal-size or uniform power superiority; classical and characteristic-function benchmarks, ablations, rounding diagnostics, and two engineering datasets mapped empirical strengths and limitations, with any heavy-tail power advantage remaining exploratory. Archived code, seeds, and replicate-level outputs support reproducibility of the paired-sample experiments.

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