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Differentially private scale testing via rank transformations and percentile modifications

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that a class of privatized Siegel–Tukey rank-sum tests for comparing two groups' spreads — RPST tests — is differentially private and asymptotically normal under the null, so p-values come from a normal table even though…

desk verdict A genuinely new class of private scale tests with solid empirics, but Theorem 3.3 is overbroad and needs a Q/n restriction; fixable and worth refereeing. read the letter →

arxiv 2507.03725 v1 pith:HTBVZW56 submitted 2025-07-04 stat.ME cs.LG

Joshua Levine , Kelly Ramsay This is my paper · ORCID
classification stat.MEcs.LG MSC 62G1062G2062G35
keywords differentialprivacyscaletestingSiegel–TukeytestranktransformationspercentilemodificationnonparametrichypothesisWald–Wolfowitztheoremsigned-rank
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

The paper introduces a class of differentially private two-sample tests for a difference in scale, the rank-transformed percentile-modified Siegel–Tukey (RPST) tests. It tries to establish that these tests are valid end to end: the privatized statistic is $(\epsilon_U+\epsilon_d,\delta)$-differentially private, its null distribution is standard normal after division by a conservatively estimated standard error, and the type I error does not exceed the nominal level. The payoff would be a practical tool for deciding whether two groups have the same spread when the data owner cannot release the group sizes or raw ranks, a testing problem the privacy literature had not carefully addressed. A second claim is that the choice of rank transformation trades noise against power: slow-growing transformations and heavy rank truncation perform best under tight privacy, fast-growing transformations under loose privacy.

What carries the argument

The load-bearing object is the RPST statistic $U_1 = \sum_{i=1}^{n_1} \psi(r_i) - \frac{n_1}{n}\sum_{i=1}^{n-Q}\psi(i)$, built from center-outward working ranks in which $Q$ central observations receive rank 0. Its global sensitivity is bounded by $GS^*(U_1) = \max\{\psi(n-Q), \, \psi(n-Q)+\psi(n-Q-1)-\bar\psi_Q\}$, which sets the Laplace noise scale; its null variance $\sigma^2(n_1,n_2,\psi,Q)$ is an explicit function of the group sizes, and Lemma A.3 shows this variance is largest when the groups are balanced. Privately underestimating $|n_1-n/2|$ therefore overestimates $\sigma$, making the test conservative. Asymptotic normality is carried by the Wald–Wolfowitz permutation central limit theorem together with Lyapunov's theorem.

What would settle it

Simulate the RPST procedure under the null with $Q=n-1$ so that only the smallest combined observation keeps a nonzero rank, with $n_1=\lfloor n/2\rfloor$ and any increasing $\psi$. For large $n$ the statistic equals one of two values depending on which group contains the minimum, so $\tilde U_1/\sigma$ concentrates on a two-point set rather than approaching $N(0,1)$; this directly contradicts the theorem's claim for $Q \in \{0,\dots,n-1\}$.

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Extended reading notes

Core claim

The central discovery is that the entire pipeline survives privatization. Observations are ranked center-outward a la Siegel–Tukey; the $Q$ most central ranks are truncated to zero; an increasing, non-negative transformation $\psi$ with $\psi(0)=0$ is applied; and Laplace noise with scale $GS^*(U_1)/\epsilon_U$ is added to the group-1 rank sum. The resulting statistic, divided by its conservative standard error $\sigma$, converges in distribution to $N(0,1)$ under the null whenever $\psi$ satisfies a general polynomial-growth condition, and the group-size imbalance $|n_1-n/2|$ is deliberately underestimated so that the reference variance is overestimated and the type I error stays at $\alpha$. The same construction, with sensitivity $2\psi(n-Q)$, upgrades the private signed-rank test.

Load-bearing premise

The normal-limit proof rests on the premise that after truncating central ranks the surviving rank values are spread out enough for the variance to stay comparable to the largest rank; when almost all ranks are truncated to zero this spread disappears and the limit need not hold.

Editorial extensions

If this is right

  • A practitioner can run a two-sample spread test with a fixed privacy budget and read p-values from a normal table, even though the group sizes $n_1$ and $n_2$ are never disclosed.
  • Under tight budgets the recommended settings are slow-growing transformations ($\tan^{-1}$, $\log$) with a large truncation fraction $q$, while under loose budgets fast-growing transformations ($\psi(r)=r$ or $r^2$) with $q\approx 0.25$ give the best power.
  • The private signed-rank test gains the same truncation-and-transformation flexibility, with sensitivity $2\psi(n-Q)$ and a normal null limit.
  • Because the sensitivity-to-standard-error ratio decays like $O((n\lambda(1-\lambda)\epsilon_U)^{-1/2})$, the privacy noise becomes negligible relative to the signal as $n$ grows, so the asymptotics guide tuning at large sample sizes.
  • The construction also supplies new non-private rank-based scale tests as a byproduct, since the RPST family had not existed in the non-private literature.

Reading between the lines

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

  • The asymptotic-normality claim as stated allows $Q \in \{0\} \cup [n-1]$; a corrected statement would need to keep $n-Q$ growing (equivalently, $Q/n$ bounded away from 1), because at $Q=n-1$ the standardized statistic is a two-point variable rather than normal.
  • The type I error control relies on the $(\epsilon,\delta)$ private group-size estimate; a pure-$\epsilon$ version would need a different construction, since the current argument uses the $\delta$-probability event that the underestimate holds.
  • The budget-allocation tradeoff could be optimized analytically from the variance and sensitivity formulas; the simulations indicate an 80/20 split toward the statistic, but the formulas make a derivation possible.
  • The same center-outward ranking with depth contours could support multivariate private scale tests, with sensitivity governed by the number of depth layers truncated.
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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 / 4 minor

Summary. The paper introduces the RPST class of differentially private two-sample scale tests. The procedure assigns center-outward Siegel–Tukey-style ranks, zeroes out the Q most central ranks, applies a nonnegative increasing transformation ψ, forms the group-one rank sum U1, adds Laplace noise calibrated to a sensitivity bound, and computes a p-value from a normal reference distribution whose variance is based on a privately underestimated group-size imbalance. The paper states a differential privacy guarantee (Theorem 3.2), an asymptotic null normality result (Theorem 3.3), a type I error bound (Lemma D.1), and an analogous private signed-rank extension (Section 5). A large simulation study compares RPST tests with a general test-of-tests framework and makes recommendations for q and ψ.

Significance. The paper fills a genuine gap: there is no dedicated differentially private two-sample scale test, and the proposal is nonparametric and computationally simple. The simulations are extensive, the code is publicly available, and the signed-rank extension is a useful byproduct. The derivations are built on standard external tools (Wald–Wolfowitz CLT, Lyapunov CLT, Laplace mechanism, composition), and the tuning parameters q and ψ are user choices rather than fitted quantities, so the theoretical development is not circular. However, Theorem 3.3 as stated overreaches; the asymptotic normality claim needs a restriction on Q or n-Q. If this is repaired, the paper would be a solid contribution to private nonparametric testing.

major comments (2)
  1. [Theorem 3.3 and Appendix A.4] Theorem 3.3 is false as stated for Q close to n. Take Q = n - 1, which is allowed because Theorem 3.3 states Q ∈ {0} ∪ [n - 1]. The working-rank vector is (ψ(1), 0, ..., 0), so U1 = ψ(1)(δ - n1/n), where δ is the indicator that the single nonzero rank falls in group 1. Under H0 and the permutation null, σ²(n1,n2,ψ,Q) = n1 n2 ψ(1)² / n² = Θ(ψ(1)²), while GS*(U1) ≥ ψ(1); hence the Laplace term Z·GS*(U1)/σ has non-vanishing scale and eU1/σ does not converge to N(0,1). The proof in Appendix A.4 asserts that Condition 3 implies Condition 4, but for this example μ_r(ψ_n,Q)/μ_2(ψ_n,Q)^{r/2} = Θ(n^{r/2 - 1}) for r > 2, so Condition 4 fails. The theorem needs an additional assumption such as n - Q → ∞ (e.g., Q/n ≤ q* < 1); the simulations use q ≤ 0.75, so this does not invalidate the empirical conclusions.
  2. [Lemma A.6] The rate O((nλ(1-λ))^{-1/2}) in Lemma A.6 is not correct for all ψ satisfying Condition 3. For ψ(i) = log(1+i), the normalized variance C(n,ψ) = n^{-1}Σ(ψ(i)/ψ(n-Q))² - (n^{-1}Σψ(i)/ψ(n-Q))² tends to zero like 1/log n because the normalized log-ranks concentrate near zero, so σ² = o(n ψ²(n-Q)) and GS*(U1)/σ ≍ √(log n / n), which is not O(n^{-1/2}). The proof's assertion that C(n,ψ) is bounded below is therefore false. Since Lemma A.6 is used in Lemma A.7 to make the Laplace term negligible, the authors should either state the correct rate for sub-polynomial ψ or prove directly that GS*(U1)/σ → 0 under Condition 3.
minor comments (4)
  1. [Section 4, parameter list] The list 'ψ(n) ∈ {tan−1(n), ln(n+1), n0.5, n, , n2}' contains a double comma after 'n'; this should be cleaned up.
  2. [Table 2, caption] In the caption of Table 2, 'respectfully' should read 'respectively'.
  3. [Theorem C.2] The definition of μ_r(yn, m) in Theorem C.2 appears to be missing a factor 1/m inside the power: it should be y_i - (1/m)Σ_{j=1}^m y_j, to be consistent with Condition 4.
  4. [Condition 3] The growth condition 'ψ(n) = O(n^k) for any k > 0' is ambiguous: if read literally as holding for every k > 0, it excludes ψ(x) = x², which the text says satisfies Condition 3. Please specify whether the intended reading is 'for some k > 0' or 'for every k > 0'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation chain: asymptotic normality comes from external Wald-Wolfowitz and Lyapunov CLTs, privacy from standard DP composition, and type I error control from a conservative variance argument; the sole self-citation is a non-load-bearing bibliographic example.

full rationale

The paper's central claims do not reduce to their inputs. Theorem 3.3 is proved in Appendix A.4 by checking the Wald-Wolfowitz condition (Condition 4), using Lemma A.6 for the vanishing Laplace-noise ratio, and applying Slutsky's theorem; the variance sigma^2 is the exact finite-sample permutation variance from Theorem A.1, and the sensitivity bound in Lemma A.4 is proved by exhaustive case analysis rather than imported from the authors' prior work. No parameter is fitted inside the theory: q and psi are user choices studied by simulation, and the private group-size construction is a DP mechanism with a proven conservative direction (Lemma D.1), not an empirical fit to the data being tested. The only self-citation, Ramsay and Chenouri (2023) at the end of Section 2.3, merely illustrates that percentile modifications were used in other rank tests and carries no weight in Theorems 3.2, 3.3, 5.1, or 5.2. One non-circular correctness gap should be noted: the proof of Theorem 3.3 asserts without adequate support that Condition 3 implies Condition 4 when n-Q is bounded; the skeptic's Q=n-1 example shows this implication can fail, so Theorem 3.3 as stated may be false in that regime. That is an invalid inference or a missing hypothesis, not a circular reduction of the conclusion to an assumption, and it does not change the circularity score.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central theory rests on standard CLTs and exchangeability under H0; the only genuinely paper-specific assumptions are the distributional conditions (Conditions 1-3) and the novel ranking convention. No new entities are postulated. Tuning parameters q and ψ are user choices, not fitted constants.

free parameters (3)
  • q (percentile modification proportion) = recommended 0.5 to 0.75 for low privacy, 0.25 for high privacy
    User-chosen tuning parameter; the paper selects recommended values by visual inspection of simulation power tables (Section 4), which is a post-hoc choice rather than a fitted constant.
  • ψ (rank transformation) = recommended tan^{-1} or ln(r+1) for low privacy; r or r^2 for high privacy
    User-chosen transformation driven by the power-sensitivity tradeoff and simulation results. The theory holds for any ψ satisfying Condition 3, so these choices are recommendations, not fitted parameters.
  • privacy budget split (ϵU, ϵd) = 80% to ϵU recommended (Appendix E.5)
    Allocation of the total privacy budget between the test statistic noise and the group-size estimate noise; determined empirically in Section 4 and Appendix E.5.
assumptions (6)
  • domain assumption Condition 1: F and G lie in the same location-scale family with equal location and are absolutely continuous
    Defines the testing problem and ensures no ties with probability 1; needed for exchangeability of ranks under H0. Introduced in Section 2.1.
  • domain assumption Condition 2: n1/n → λ ∈ (0,1)
    Ensures both group sizes grow proportionally; standard in two-sample rank testing. Section 3.2.
  • ad hoc to paper Condition 3: ψ continuous, strictly increasing, ψ(0)=0, ψ(n)=O(n^k) for some k > 0
    A sufficient condition on the rank transformation for asymptotic normality of the rank sum; the paper notes Condition 4 is a weaker but more opaque version. Section 3.2 and Appendix A.4.
  • domain assumption Exchangeability of group labels under H0
    Under H0 with i.i.d. samples, the combined sample order statistics are independent of the group label permutation; this justifies modeling the group 1 rank sum as a simple random sample without replacement. Used throughout the proofs, especially in the application of the Wald-Wolfowitz theorem.
  • standard math Wald-Wolfowitz central limit theorem (Theorem C.2) and Lyapunov CLT (Theorem C.1)
    Used to prove asymptotic normality of sums without replacement and of the signed-rank statistic; quoted from Wald and Wolfowitz [23] and Pratt and Gibbons [18], Appendix C.
  • standard math Definitions and composition properties of differential privacy (Definitions 2.1, Propositions 2.1 and 2.2)
    Backbone of the privacy guarantees; standard results cited from Dwork et al.

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Cite this review

Pith. "Pith review of Differentially private scale testing via rank transformations and percentile modifications." pith.science (2026). https://pith.science/paper/HTBVZW56

@misc{pith2026250703725,
  author       = {Pith},
  title        = {Pith review of: Differentially private scale testing via rank transformations and percentile modifications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HTBVZW56}},
  note         = {Machine review of arXiv:2507.03725}
}
read the original abstract

We develop a class of differentially private two-sample scale tests, called the rank-transformed percentile-modified Siegel--Tukey tests, or RPST tests. These RPST tests are inspired both by recent differentially private extensions of some common rank tests and some older modifications to non-private rank tests. We present the asymptotic distribution of the RPST test statistic under the null hypothesis, under a very general condition on the rank transformation. We also prove RPST tests are differentially private, and that their type I error does not exceed the given level. We uncover that the growth rate of the rank transformation presents a tradeoff between power and sensitivity. We do extensive simulations to investigate the effects of the tuning parameters and compare to a general private testing framework. Lastly, we show that our techniques can also be used to improve the differentially private signed-rank test.

Figures

Figures reproduced from arXiv: 2507.03725 by the authors.

Figure 1
Figure 1. An example of the ranking procedure for the RPST test, applied to a sample of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Left: The function Πn(ψ, Q, 1) as a function of q = Q/n for different ψ at n = 100, with n1 = n2. We see that for larger values of q and slow growing ψ is optimal, while the opposite is true for small values of q. Overall, we see that larger values of q with slow growing ψ seems to work well. Right: Estimates of βn(ψ, Q) when F = N (0, 1) and G = N (0, 9) via Monte Carlo simulation, as a function of q = Q/n for diff… view at source ↗
Figure 3
Figure 3. Empirical power of the RPST test with different group sizes under normally [PITH_FULL_IMAGE:figures/full_fig_p052_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Empirical power of the RPST test with different budget allocations under normally [PITH_FULL_IMAGE:figures/full_fig_p053_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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  16. [24]

    E(Sk) = k n Pn i=1 xi

  17. [25]

    Var(Sk) = k n Pn i=1 x2 i + 2 k(k−1) n(n−1) Pn i=1 Pn j=i+1 xixj − k n Pn i=1 xi 2 . Proof. We begin with assertion 1. Let A = ( A1, A2, . . . , Ak) be a random sample from (x1, . . . , xn). For each i ∈ {1, 2, . . . , n}, let δi = 1{xi ∈ A}. Then, E(Sk) = E kX i=1 Ai ! = E nX...

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    raw rank

    V is decreasing in k − n 2 . This is what we wanted to show. A.3 The privacy of the test We now, prove the privacy of the test, which follows mainly from the following lemma: Lemma A.4. It holds that for all Q ∈ {0, . . . , n}, and all increasing, non-negative real functions ψ...

  20. [30]

    This means that: GS(U1) ≤ | −r(p, n) + r(q, n) − |b2||

    The reduction of the raw ranks of the data points in region A2 of database x by one apiece, which decreases U1 by |b2| (as working ranks decrease away from the centre). This means that: GS(U1) ≤ | −r(p, n) + r(q, n) − |b2||. Since r(1, n) = ψ(1) ≤ ||b2| + r(p, n)| ≤r(h, n) − r...

  21. [32]

    The addition of the row (whose raw rank becomes q) into database x′, which increases U1 by r(q, n)

  22. [34]

    This means that: GS(U1) ≤ | −r(p, n) + r(q, n) + |b3| − |b2||

    The reduction of the raw ranks of the data points in region A2 of database x by one apiece, which decreases U1 by |b2| (as working ranks decrease away from the centre). This means that: GS(U1) ≤ | −r(p, n) + r(q, n) + |b3| − |b2||. Since 0 ≤ ||b2 + r(p, n)| ≤r(h, n) − r(1, n) ...

  23. [36]

    The reduction of the raw ranks of the data points in region A2 of database x by one apiece, which decreases U1 by |b2| (as working ranks decrease away from the centre)

  24. [37]

    23 This means that: GS(U1) ≤ −r(p, n) − |b2| + 1 n nX i=1 ψ(i)

    The removal of a row from group 1, which increases U1 by 1 n Pn i=1 ψ(i). 23 This means that: GS(U1) ≤ −r(p, n) − |b2| + 1 n nX i=1 ψ(i) . Since ψ(1) ≤ ||b2| + r(p, n)| ≤r(h, n) − r(1, n) = ψ(n) − ψ(1), and Pn i=1 ψ(i) ≤ ψ(n), we have that GS( U1) ≤ ψ(n) − ψ(1). By a similar a...

  25. [38]

    The removal of the row with raw rank p from database x, which (by itself) decreases U1 by r(p, n)

  26. [39]

    The removal of a row from group 1, which increases U1 by 1 n Pn i=1 ψ(i)

  27. [40]

    The reduction of the raw ranks of the data points in region A3 of database x by one apiece, which increases U1 by |b3| (as working ranks increase toward the centre)

  28. [41]

    raw rank

    The reduction of the raw ranks of the data points in region A2 of database x by one apiece, which decreases U1 by |b2| (as working ranks decrease away from the centre). This means that: GS(U1) ≤ −r(p, n) + 1 n nX i=1 ψ(i) + |b3| − |b2| . To make further progress, we will utili...

  29. [42]

    A1, consisting of the rows with ranks not exceeding rj; 29

  30. [43]

    A2, consisting of the rows with ranks exceeding rj but with ai < a ′ j, where a ′ j is the absolute difference of row j of database w; and,

  31. [44]

    Before proceeding further, we will define some more notation

    A3, consisting of the rows with ai > a ′ j. Before proceeding further, we will define some more notation. Let ∆ j be the change in the signed rank of row j when transitioning from v to w. In addition, for i = 1 , 2, 3, let ∆Ai be the change in the sum of the signed ranks of al...

  32. [45]

    ∂πα ∂m is positive for all m ∈ 1, n 2

  33. [46]

    Let us start with item 1

    πα(0; n) < πα(1; n). Let us start with item 1. We note that n 2 − m is positive for all m ∈ 1, n 2 . Consequently, ∂πα ∂m is positive whenever erf −1 (1 − α)p 6m(n − m) < 1. We may also note that m(n − m) is increasing in m for 0 < m < n/2. Hence, the left-hand side of the abo...

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