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arxiv: 2604.15696 · v1 · submitted 2026-04-17 · 📊 stat.ME · math.PR

Testing and estimation of the index of stability of univariate and bivariate symmetric α-stable distributions via modified Greenwood statistic

Katarzyna Skowronek , Marek Arendarczyk , Anna K. Panorska , Tomasz J. Kozubowski , Agnieszka Wy{\l}oma\'nska This is my paper

Pith reviewed 2026-05-10 09:04 UTC · model grok-4.3

classification 📊 stat.ME math.PR
keywords symmetric alpha-stable distributionsGreenwood statisticstability index estimationbivariate testingheavy-tailed distributionssub-Gaussian casestatistical testing
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The pith

The modified Greenwood statistic enables testing for bivariate symmetric alpha-stable distributions and estimation of the stability index in univariate cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a methodology using a modified Greenwood statistic to test whether univariate or bivariate data follows a symmetric alpha-stable distribution and to estimate the stability parameter. The statistic, originally for positive variables, is adapted for symmetric data and extended to joint bivariate samples with focus on the sub-Gaussian case. A sympathetic reader would care because alpha-stable laws model heavy-tailed phenomena in finance, signals, and physics, yet distinguishing them from Gaussian distributions becomes difficult as the index approaches 2. The authors derive probabilistic properties, propose test statistics for the bivariate setting, adapt the approach for estimation in the univariate setting, and demonstrate through simulations that it outperforms classical methods while handling real data examples.

Core claim

We extend the modified Greenwood statistic to a bivariate setting and examine its probabilistic properties within the class of alpha-stable distributions, with a focus on the sub-Gaussian case. We introduce a novel testing approach that considers two variations of the modified Greenwood statistic as test statistics for the bivariate case. In the univariate setting, we adapt the proposed testing methodology for estimating the stability index. The simulation studies presented demonstrate that our proposed methodology outperforms classical approaches previously used in this context and serves as an effective tool for distinguishing between Gaussian and alpha-stable distributions with a index of

What carries the argument

The modified Greenwood statistic for symmetric distributions, extended to bivariate samples to serve as a test statistic and basis for stability index estimation.

If this is right

  • Enables construction of tests for whether bivariate observations follow a joint symmetric alpha-stable law.
  • Provides a direct estimator for the stability index from univariate samples.
  • Yields improved ability to separate alpha-stable laws from Gaussian ones when the index is close to 2.
  • Supplies explicit test statistics based on two variants of the modified Greenwood statistic for the bivariate setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bivariate extension opens the possibility of joint modeling of heavy-tailed pairs of observations in applications such as portfolio risk or paired sensor data.
  • Because the method focuses on the sub-Gaussian regime, it may serve as a template for similar statistics on other symmetric heavy-tailed families.
  • Practical deployment would require checking robustness when the data only approximately follows the symmetric alpha-stable assumption.

Load-bearing premise

Data samples are drawn from symmetric alpha-stable distributions and the modified Greenwood statistic has the examined probabilistic properties within this class, particularly in the sub-Gaussian case.

What would settle it

A simulation experiment in which the proposed bivariate tests or univariate estimator fails to outperform classical methods when distinguishing Gaussian data from alpha-stable data with stability index near 2 would falsify the performance claim.

Figures

Figures reproduced from arXiv: 2604.15696 by Agnieszka Wy{\l}oma\'nska, Anna K. Panorska, Katarzyna Skowronek, Marek Arendarczyk, Tomasz J. Kozubowski.

Figure 1
Figure 1. Figure 1: The CDF (left panel) and the tail of the distributio [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the power curves of a test based on [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the power of a test based on [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the power of tests for for bivariate G [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The residuals of analyzed financial dataset after u [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

We propose a testing and estimation methodology for univariate and bivariate symmatric $\alpha$-stable distributions using a modified version of the Greenwood statistic. Originally designed for positive-valued random variables, the Greenwood statistic, and its modified version tailored for symmetric distributions, have been predominantly applied to univariate random samples. In this paper, we extend the modified Greenwood statistic to a bivariate setting and examine its probabilistic properties within the class of $\alpha$-stable distributions, with a focus on the sub-Gaussian case. Additionally, we introduce a novel testing approach that considers two variations of the modified Greenwood statistic as test statistics for the bivariate case. In the univariate setting, we adapt the proposed testing methodology for estimating the stability index. The simulation studies presented demonstrate that our proposed methodology outperforms classical approaches previously used in this context and serves as an effective tool for distinguishing between Gaussian and $\alpha$-stable distributions with a stability index close to 2. The theoretical and simulation results are further illustrated with practical data examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper proposes extending the modified Greenwood statistic to bivariate symmetric α-stable distributions and examines its probabilistic properties (with focus on the sub-Gaussian case). It develops testing procedures using two variants of the statistic for the bivariate setting, adapts the approach for estimating the stability index α in the univariate case, and reports simulation results claiming outperformance over classical methods, particularly for distinguishing Gaussian (α=2) from α-stable laws with α close to 2. Theoretical results and simulations are illustrated on practical data examples.

Significance. If the probabilistic properties are shown to hold uniformly for α ∈ (1,2] and the simulation comparisons are reproducible with full controls, the work could provide a useful addition to the limited set of tools for inference on near-Gaussian stable laws, which arise in financial modeling and signal processing. The bivariate extension and data examples add practical value, though the reliance on simulations for the key distinction claim limits the strength of the contribution.

major comments (3)
  1. [Section 3] Section 3 (probabilistic properties): The derivations and limiting results for the modified Greenwood statistic are presented with explicit focus on the sub-Gaussian case. However, the central claim requires reliable behavior for α close to but strictly less than 2, where the tails are heavier and quadratic functionals may lack the same integrability. No continuity argument, uniform integrability result, or separate derivation for α ∈ (1,2) is provided to justify extrapolation of the test statistic's distribution or moments to the near-Gaussian alternatives used in the simulations.
  2. [Section 5] Section 5 (simulation studies): The reported outperformance in distinguishing α=2 from α near 2 and in estimation accuracy lacks accompanying standard errors, confidence intervals, or details on the number of Monte Carlo replications, exact grid of α values tested, and data-generation parameters. Without these, it is impossible to assess whether the claimed superiority is statistically significant or sensitive to simulation design choices.
  3. [Section 4] Section 4 (bivariate testing): The two proposed test statistics for the bivariate case are introduced without a clear power analysis or comparison to existing bivariate stable tests under local alternatives near α=2. This leaves the practical advantage of the bivariate extension unsubstantiated beyond the univariate estimation results.
minor comments (3)
  1. [Abstract] The abstract contains a typographical error ('symmatric' instead of 'symmetric').
  2. [Section 2] Notation for the modified Greenwood statistic in the bivariate setting should be introduced with an explicit definition before its use in the testing procedures.
  3. [Discussion] The paper would benefit from a brief discussion of how the proposed estimator behaves when the underlying distribution deviates from exact symmetry, even if the main focus is symmetric α-stable laws.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and strengthen the presentation of our results. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (probabilistic properties): The derivations and limiting results for the modified Greenwood statistic are presented with explicit focus on the sub-Gaussian case. However, the central claim requires reliable behavior for α close to but strictly less than 2, where the tails are heavier and quadratic functionals may lack the same integrability. No continuity argument, uniform integrability result, or separate derivation for α ∈ (1,2) is provided to justify extrapolation of the test statistic's distribution or moments to the near-Gaussian alternatives used in the simulations.

    Authors: The derivations in Section 3 are deliberately restricted to the sub-Gaussian case because this setting yields explicit limiting distributions via the properties of multivariate Gaussians and the associated quadratic forms. For the univariate estimation procedure and the simulation study of behavior near α = 2, we rely on the empirical performance shown in Section 5. We agree that a formal bridge between the sub-Gaussian limit and α ∈ (1,2) is not supplied. In the revision we will insert a short remark in Section 3 invoking the continuous mapping theorem together with the fact that symmetric α-stable laws converge weakly to the Gaussian law as α → 2^−; under the moment conditions already verified for the Greenwood statistic this implies convergence of the relevant functionals. This addition will be limited to a remark rather than a full re-derivation. revision: partial

  2. Referee: [Section 5] Section 5 (simulation studies): The reported outperformance in distinguishing α=2 from α near 2 and in estimation accuracy lacks accompanying standard errors, confidence intervals, or details on the number of Monte Carlo replications, exact grid of α values tested, and data-generation parameters. Without these, it is impossible to assess whether the claimed superiority is statistically significant or sensitive to simulation design choices.

    Authors: We accept that the simulation section must be made fully reproducible. The revised manuscript will report: 10 000 Monte Carlo replications for each configuration, the precise grid α = 1.01, 1.05, …, 1.95, 2.00, Monte Carlo standard errors and 95 % confidence intervals for every rejection rate and MSE value, and the exact data-generation protocol (Chambers–Mallows–Stuck algorithm with fixed random seeds). These additions will allow readers to judge both statistical significance and sensitivity to design choices. revision: yes

  3. Referee: [Section 4] Section 4 (bivariate testing): The two proposed test statistics for the bivariate case are introduced without a clear power analysis or comparison to existing bivariate stable tests under local alternatives near α=2. This leaves the practical advantage of the bivariate extension unsubstantiated beyond the univariate estimation results.

    Authors: Section 4 presents the bivariate extension together with the null limiting distributions under the sub-Gaussian assumption; the data examples illustrate its use. We acknowledge that a dedicated power comparison against existing bivariate procedures (e.g., characteristic-function or likelihood-based tests) under local alternatives is absent. In the revision we will add a concise simulation subsection (or appendix) that evaluates empirical power of the two proposed statistics for α = 1.90, 1.95, 1.99 against a benchmark empirical-characteristic-function test, using the same Monte Carlo settings as Section 5. This will directly address the practical advantage of the bivariate extension. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation and validation are independent

full rationale

The paper extends the modified Greenwood statistic to the bivariate case for symmetric alpha-stable laws, examines its probabilistic properties (with explicit focus on the sub-Gaussian regime), introduces corresponding test statistics and an estimation procedure for the stability index, and reports simulation results showing outperformance versus classical methods. No load-bearing step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or self-citation chain by the paper's own equations. The simulation-based validation constitutes an external check rather than a tautology, rendering the central claims self-contained against the stated assumptions and benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the central claim rests on standard properties of symmetric alpha-stable distributions and the probabilistic behavior of the Greenwood statistic extension; no free parameters or invented entities are introduced in the summary.

axioms (1)
  • domain assumption Symmetric alpha-stable distributions possess well-defined probabilistic properties that allow extension of the modified Greenwood statistic, especially in the sub-Gaussian case.
    Invoked when examining properties and applying the statistic to testing and estimation.

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