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REVIEW 2 major objections 5 minor 36 references

Novel computational approaches for ratio distributions with an application to Hake's ratio in effect size measurement

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that two numerical methods—1D double-exponential quadrature of a Mellin convolution and 2D vectorized Broda–Khan characteristic-function inversion—compute ratio-distribution densities with accuracy matching the best…

desk verdict The 1D Mellin–DE method is the real, well-validated contribution; the 2D Broda–Kan part works empirically but the stated theorem's convergence condition is not satisfied for the pilot cases, so the paper needs a caveat or a repaired argument. read the letter →

arxiv 2411.12938 v2 pith:OUIQV77Z submitted 2024-11-20 stat.CO physics.data-an

Jozef Hanč , Martina Hančová , Dominik Borovský This is my paper · ORCID
classification stat.COphysics.data-an MSC 62E1565D30
keywords RatiostatisticsExactprobabilitydistributionsIntegraltransformsDoubleexponentialquadratureHakenormalizedgainEffectsizeCharacteristicfunctioninversion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that two numerical techniques can compute the probability density of a ratio of random variables quickly and accurately enough for practical statistics. The first technique applies one-dimensional double-exponential quadrature to the Mellin convolution integral for the ratio's PDF; the second uses a fully vectorized two-dimensional Broda–Khan inversion of characteristic functions. Both are tested on Hake's normalized gain, the standard effect-size measure in physics education, which is the ratio of two correlated normal variables. The authors claim the Python implementation of the first method matches the accuracy of the established PaCAL package and reaches speeds comparable to C code, while the second method supplies the first functional reproducible implementation of Broda–Khan inversion. If these claims hold, researchers can obtain exact ratio-distribution calculations in open tools instead of relying on Monte Carlo or normal approximations.

What carries the argument

The paper's argument rides on two integral representations of a ratio's density. The first is the Mellin convolution for independent variables, $f_T(t)=\int_{-\infty}^{\infty} f_1(xt)f_2(x)\,|x|\,dx$, evaluated by 1D double-exponential quadrature using a $\sinh$-$\sinh$ transformation; the DE transformation makes the trapezoidal rule converge exponentially for a broad class of integrands. The second is the Broda–Khan inversion formula, $f_T(x)=\frac{1}{\pi^2}\int_0^\infty\int_{-\infty}^\infty \Re\left[\varphi_{X_1}(s)\,\varphi'_{X_2}(-t-xs)/t\right]\,ds\,dt$, evaluated by a vectorized 2D trapezoidal or Chebyshev quadrature. The Broda–Khan formula is what lets the method work from characteristic functions alone, without requiring known PDFs or independence of the ratio constituents.

What would settle it

Evaluate the Broda–Khan inversion on a ratio of heavy-tailed variables, for example $X_1\sim N(0,1)$ and $X_2\sim t_3$, by comparing the 2D method's output against a high-precision evaluation of the exact Mellin convolution integral. If the error exceeds the reported tolerance or the absolute-convergence condition stated in Theorem 6 is not satisfied, the paper's claim that the 2D method is broadly applicable and reliable would be refuted.

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Extended reading notes

Core claim

On its own terms, the paper establishes that two numerical routes can replace black-box Monte Carlo for ratio-distribution calculations. The 1D route evaluates the Mellin convolution integral $f_T(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-((xt-a)^2+(x-b)^2)/2}|x|\,dx$ with double-exponential quadrature, reaching maximum absolute errors around $10^{-16}$ in sub-millisecond times when compiled with Numba and accelerated with barycentric interpolation. The 2D route evaluates the Broda–Khan inversion integral with a vectorized trapezoidal or Chebyshev quadrature; it is slower by roughly two orders of magnitude but still practically usable, and the authors state it is the first functional reproducible implementation of that inversion. The pilot study on Hake's normalized gain, a ratio of correlated normal variables, shows stable runtimes and errors across four shape regimes, and a preliminary $N(0,1)/\chi^2(5)$ example indicates the 2D method also works beyond normal constituents. The claim is therefore that these are fast, accurate, and reliable open-tool methods for ratio distributions, not just for this application.

Load-bearing premise

The second method assumes that the double integral defining the density settles to a finite value before the numerical truncation happens, but the paper never verifies this assumption directly for the cases it runs.

Editorial extensions

If this is right

  • For Hake's normalized gain, researchers can now compute exact PDF values numerically in milliseconds instead of relying on Monte Carlo or normal approximations, using the paper's notebooks and code.
  • The 1D double-exponential Mellin method, when compiled with Numba, reaches speeds comparable to the 20-year-old C code and to PaCAL's Cython implementation, while exposing a single accuracy parameter that trades speed against error.
  • The 2D Broda–Khan implementation requires only characteristic functions, so it can handle non-normal and negative-valued ratio constituents without assuming independence.
  • Across the four tested shape regimes of Hake's ratio, both methods kept maximum absolute errors stable and runtimes within a 5% coefficient of variation.
  • Preliminary results for $X_1\sim N(0,1)$ over $X_2\sim\chi^2(5)$ indicate the Broda–Khan method extends to non-normal ratios, with agreement against a PDF-based approach.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the absolute-convergence condition behind the Broda–Khan formula is checked case by case, the vectorized 2D implementation could give metrology a practical default for uncertainty propagation when the measurement model's characteristic function is known but its density is not.
  • The 1D DE–Mellin route should transfer directly to other standardized effect-size ratios where the constituents' PDFs are known; a head-to-head run against the same analytic formulas used here would be a cheap validation.
  • A timing comparison that includes PaCAL's roughly one-second initialization would clarify whether the reported speed parity holds in interactive use, where overhead matters as much as steady-state runtimes.
  • The pilot study's four parameter pairs cover four qualitative shapes of the ratio distribution, so the natural next step is a systematic parameter sweep to map where each method's quadrature parameters need tuning.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes two numerical methods for computing the density of a ratio of random variables: a one-dimensional double-exponential (DE) quadrature of the Mellin convolution integral, optionally combined with barycentric interpolation, and a two-dimensional vectorized implementation of the Broda--Kan characteristic-function inversion. The methods are applied to Hake's normalized gain, modeled as a ratio of normal random variables, and are benchmarked against the analytic Hinkley--Marsaglia density evaluated at quadruple precision. The pilot study reports runtimes, speedups, and maximum absolute errors for four representative (a,b) parameter pairs, and the authors provide Jupyter notebooks and code on GitHub.

Significance. The external validation strategy is a clear strength: the reference values come from an independent analytic formula computed at quadruple precision and cross-checked with Arb, rather than from the methods' own outputs, so the reported errors for the 1D DE Mellin-convolution method are meaningful. For the 1D method, the paper convincingly demonstrates that a Numba-based Python DE implementation can match PaCAL's accuracy and approach C-like speeds, and the reproducibility artifacts are a substantial asset. The 2D Broda--Kan implementation addresses a real gap, since no widely available vectorized implementation exists. However, the theoretical justification of the 2D method is not currently secure: the stated inversion theorem requires absolute convergence, a condition that is not verified and is in fact false for the normal cases tested, as discussed below. If that gap is repaired, the paper would be a useful contribution to computational statistics and metrology.

major comments (2)
  1. [Appendix B, Theorem 6 and Eq. (3.13)] The theorem is invoked as the theoretical basis for the 2D method, but its hypothesis of absolute convergence is not satisfied for the pilot's normal cases. For independent X1 ~ N(a,1) and X2 ~ N(b,1), the integrand in (B4) has a non-removable 1/t singularity at t=0 whose coefficient, Re[phi_{X1}(s)(ib+xs) exp(-ibxs - x^2 s^2 / 2)], is odd in s, so the double integral converges only conditionally after the s-integration cancels the singularity; the integral of the absolute value diverges logarithmically. The manuscript neither verifies absolute convergence nor states an improper/principal-value or iterated-integral interpretation. Consequently, the reported accuracy of the 2D method is not logically connected to the theorem as stated. Please add a rigorous treatment of the conditional convergence (or replace (B4) with an equivalent difference form over t>0 with a removable singularity), and state explicitly which quadrature interpretation Algorithm 1 implements.
  2. [Appendix C, Algorithm 1] The pseudocode is not self-contained enough to support the reproducibility claim. It does not state the quadrature limits, the step sizes, or the fact that the t2 summation is restricted to positive half-offset points while the t1 summation is symmetric; those choices are exactly what make the conditionally convergent integral in (3.13) well defined. As written, the algorithm is a bare trapezoidal sum of a non-absolutely convergent integral, and a reader cannot determine from the paper alone how N and h were chosen or how convergence was monitored. Please provide the complete quadrature rule and a concrete convergence criterion, and clarify how the singularity cancellation is handled.
minor comments (5)
  1. [Section 5] The paragraph beginning 'The method's speed, accuracy, and reliability were tested...' appears twice in the conclusions with only minor wording differences; please remove the duplicate.
  2. [Section 4.1] The sentence 'The only commercial is MATLAB' is grammatically incomplete; it should read 'The only commercial tool is MATLAB.'
  3. [Sections 3.1 and 4.2] The software name is spelled both 'PaCAL' and 'PACAL'; please standardize to the package's official capitalization.
  4. [Eq. (2.6)] The notation in Eq. (2.6) uses delta_2 both for the coefficient of variation of X2 and for the sum delta_1^2 + delta_2^2 in the variance expression; please introduce a distinct symbol for the combined quantity to avoid confusion.
  5. [Section 4.1 and Table 2] In Table 2, the accuracy column reports values such as '3e-16' and '1e-02' without stating that these are maximum absolute errors; please add a note that all accuracy entries are epsilon_max as defined in Section 4.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: numerical methods are validated against an independent analytic formula; self-citations are tool provenance, not load-bearing.

full rationale

The paper's central claims are that its 1D DE Mellin-convolution implementation and its 2D vectorized Broda-Kan inversion compute ratio PDFs accurately and quickly. Neither claim reduces to its own inputs. Accuracy is assessed by comparing against the external closed-form Hinkley/Marsaglia/Pham-Gia formula (2.3) with (2.5), computed independently in SageMath with quadruple precision and cross-checked against Arb; no parameter of the proposed methods is fitted to the target PDF. The pilot cases (a,b) are fixed inputs, not fitted outputs. Self-citations (Gajdos et al. 2021, Hancova et al. 2022) are used only to attribute the provenance of DE-quadrature code and prior usage, not as evidence for the numerical benchmark results. The Broda-Kan inversion theorem is cited to an external source (Broda and Kan 2016). The possible convergence-condition concerns about Theorem 6 and Appendix C are correctness risks about the validity of the integral representation, not circularity: they do not make the derivation equivalent to its inputs. Accordingly no circular step is identified, and the score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numbers were fitted to data to produce the central result. The (a,b) test cases and quadrature tolerances are experiment settings, not free parameters. The ledger lists the classical theorems used as background and the convergence and normality assumptions that the methods inherit. No new entities are postulated.

assumptions (6)
  • standard math Mellin convolution formula for the PDF of a ratio of independent random variables (Theorem 3, Eq. 3.8).
    Defines the 1D DE quadrature target. It is a classical result, but the method inherits its independence and known-PDF conditions.
  • standard math Closed-form PDF of the ratio of bivariate normal random variables in terms of Kummer 1F1 or erf (Theorems 1 and 2, Eqs. 2.2 to 2.5).
    Used as the external benchmark for all numerical results. The ground truth is only as good as this formula.
  • standard math Gil-Pelaez inversion formulas for PDF and CDF from a characteristic function (Eq. 3.10).
    Background for the CF inversion route. Standard Fourier inversion result.
  • domain assumption Broda-Kan ratio inversion theorem requires finite mean, absolutely integrable CF, no atom at X - rY = 0, and absolute convergence of the integrals (Theorem 6, Appendix B, Eqs. B3 and B4).
    The paper uses this theorem for the 2D method without proving the conditions for the four tested (a,b) pairs or the chi-square example. Convergence is only checked through empirical error monitors.
  • domain assumption Scores in Hake's ratio are normal or approximately normal via the central limit theorem (Proposition 4 and Remark 8, Appendix B).
    Justifies treating Hake's normalized gain as a ratio of correlated normal random variables. The paper acknowledges that n > 30 or stronger conditions may be needed.
  • standard math Double-exponential quadrature converges exponentially for the integrands considered, with optimality results as in Sugihara (1997).
    Underlies the speed and accuracy claims for the 1D Mellin convolution. Accepted numerical analysis, but it depends on integrand regularity.

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Cite this review

Pith. "Pith review of Novel computational approaches for ratio distributions with an application to Hake's ratio in effect size measurement." pith.science (2026). https://pith.science/paper/OUIQV77Z

@misc{pith2026241112938,
  author       = {Pith},
  title        = {Pith review of: Novel computational approaches for ratio distributions with an application to Hake's ratio in effect size measurement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUIQV77Z}},
  note         = {Machine review of arXiv:2411.12938}
}
read the original abstract

Ratio statistics and distributions are fundamental in various disciplines, including linear regression, metrology, nuclear physics, operations research, econometrics, biostatistics, genetics, and engineering. In this work, we introduce two novel computational approaches for evaluating ratio distributions using open data science tools and modern numerical quadratures. The first approach employs 1D double exponential quadrature of the Mellin convolution with/without barycentric interpolation, which is a very fast and efficient quadrature technique. The second approach utilizes 2D vectorized Broda-Khan numerical inversion of characteristic functions. It offers broader applicability by not requiring knowledge of PDFs or the independence of ratio constituents. The pilot numerical study, conducted in the context of Hake's ratio - a widely used measure of effect size and educational effectiveness in physics education - demonstrates the proposed methods' speed, accuracy, and reliability. The analytical and numerical explorations also provide more clarifying insight into the theoretical and empirical properties of Hake's ratio distribution. The proposed methods appear promising in a robust framework for fast and exact ratio distribution computations beyond normal random variables, with potential applications in multidimensional statistics and uncertainty analysis in metrology, where precise and reliable data handling is essential.

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