{"id":"b7a89a7c-24ae-4d2b-8ecb-e009b636f198","arxiv_id":"2507.03725","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"New differentially private rank-based tests for two-sample scale differences achieve controlled type I error and often beat generic private testing frameworks in power.","lead":"This paper introduces new statistical tests that compare the spread of two groups under differential privacy, the standard formal guarantee that individual data points stay hidden. The tests add tunable rank transformations and noise reduction tricks that make them more powerful than generic privacy-protecting test frameworks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.3 fails as stated for Q near n: with Q=n-1 the working ranks are a single nonzero entry, so Condition 4's moment ratio diverges and Laplace noise does not vanish relative to sigma; the theorem needs n-Q to grow, e.g. Q/n <= q* < 1.","rationale":"The paper proposes a genuinely new class of private rank-based scale tests and supports them with extensive simulations; the recommended settings (q <= 0.75) appear sound and the empirical sizes are near nominal. The load-bearing issue is narrow but real: Theorem 3.3 is stated for all Q in {0} union [n-1], yet its proof (Appendix A.4) requires Condition 4, and the assertion that Condition 3 implies Condition 4 is false when n-Q is bounded. At Q=n-1, the rank vector has a single nonzero value: mu_2 = Theta(1/n) and mu_4 = Theta(1/n), so mu_4/mu_2^2 = Theta(n); and GS*/sigma = Theta(sqrt(n)), so the Laplace noise dominates. The theorem therefore fails as stated, and the abstract's unconditional type I error guarantee is too strong. This is an internal inconsistency in a theorem and its proof, not a disagreement with established consensus. It is fixable by restricting Q, e.g., Q/n <= q* < 1 or requiring n-Q to grow, which matches the paper's own simulations. The reader's weakest_assumption identifies exactly this gap; I agree. My concern does not move the verdict: CONDITIONAL remains appropriate, with the authors required to correct the theorem's statement and proof, adjust Lemma A.6, and soften the abstract.","tokens_in":36701,"tokens_out":13511,"duration_ms":137882,"concrete_test":"For Q=n-1 and psi(x)=x, explicitly compute GS*(U1)=psi(n-Q)=1 and sigma^2(n1,n2,psi,Q)=n*lambda*(1-lambda)*(bar{psi^2}_Q - (bar{psi}_Q)^2) = (n-1)/n^2 approx 1/n; the ratio GS*/sigma scales as sqrt(n), not O(n^{-1/2}). Since Lemma A.6's bound is the step that makes the Laplace noise vanish in Theorem 3.3's proof, this failure confirms the theorem's stated range Q in {0} union [n-1] is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim is Theorem 3.3 (Appendix A.4): for every Q in {0} union [n-1], Conditions 1-3 give eU1/sigma(en1,en2,psi,Q) converging to N(0,1). Its proof asserts Condition 3 implies Condition 4, the Wald-Wolfowitz moment condition mu_r(psi_n,Q,n)/mu_2(psi_n,Q,n)^{r/2} = O(1) for all r>2. This implication is false when n-Q is bounded. For Q=n-1, the transformed-rank vector is (psi(1),0,...,0), so mu_2 = (n-1)/n^2 * psi(1)^2 and mu_r = Theta(psi(1)^r/n); hence mu_r/mu_2^{r/2} = Theta(n^{r/2-1}) -> infinity for r>2. The same regime breaks Lemma A.6: sigma = Theta(psi(1)/sqrt(n)) but GS*(U1) = psi(n-Q) = psi(1), so GS*/sigma = Theta(sqrt(n)) instead of O(n^{-1/2}). The additive Laplace noise therefore does not vanish after standardization, and eU1/sigma does not converge to N(0,1). A corrected theorem must require n-Q -> infinity, e.g. Q/n <= q* < 1; this covers the paper's own simulations (q <= 0.75). Without this restriction, the abstract's unconditional type I error claim is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":37020,"tokens_out":12501,"duration_ms":145902,"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":[{"comment":"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.","section":"Theorem 3.3 and Appendix A.4"},{"comment":"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.","section":"Lemma A.6"}],"minor_comments":[{"comment":"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.","section":"Section 4, parameter list"},{"comment":"In the caption of Table 2, 'respectfully' should read 'respectively'.","section":"Table 2, caption"},{"comment":"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.","section":"Theorem C.2"},{"comment":"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'.","section":"Condition 3"}],"recommendation":"major_revision","confidential_remarks":"I found no citation or novelty concerns; the main barrier is the overstatement of Theorem 3.3 and the incorrect rate in Lemma A.6. Both are local and fixable by adding a condition such as n-Q → ∞ and by correcting the sensitivity-to-variance rate for slowly growing ψ, so I would support acceptance after a careful revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely new class of DP scale tests—no one had done a private Siegel–Tukey-type test—and the empirical work is solid. But the central theorem is overclaimed: as stated, Theorem 3.3 fails for Q close to n, and a key lemma's proof has a wrong rate for slowly growing transformations. Both are fixable, and the paper's own simulations stay in the safe regime.\n\nThe new stuff is real: combining Gastwirth's percentile modification with a rank transformation, an extremes-inward ranking scheme, and the Couch et al. group-size estimate into a private scale test. The asymptotic normality result under a general condition on ψ, the sensitivity-power tradeoff, and the extension to the signed-rank test are all worthwhile. The simulations are extensive, cover several distributions and privacy budgets, and show the specialized test beating the 'test of tests' framework for n≥500. Code is available. All of that is to the good.\n\nThe soft spots are specific and serious but not fatal. Theorem 3.3 claims convergence for every Q ∈ {0} ∪ [n−1]. The proof of Condition 3 ⇒ Condition 4 collapses when n−Q is bounded: for Q=n−1 the transformed-rank vector is a single nonzero entry, the moment ratio μ_r/μ_2^{r/2} diverges, and GS*/σ does not go to zero, so the Laplace noise does not vanish. The theorem needs n−Q to grow; the natural fix is Q/n ≤ q* < 1, which is exactly the range used in the simulations (q ≤ 0.75). Lemma A.6 also asserts C(n,ψ) is bounded below, which fails for ψ(n)=log(n+1) where C → 0; the rate is then O(log n/√n), not O(n^{−1/2}). That is a proof gap rather than a false conclusion, but it should be corrected. The abstract's unconditional type I error claim should be softened to match the corrected theorem.\n\nThe paper is clearly written, the derivations are parameter-free, and the authors engage honestly with the literature—the appendix even corrects an oversight in Couch et al. I would send this to a serious referee. The revisions are mechanical: fix the theorem statement, patch the lemma, and qualify the abstract.","headline":"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.","tokens_in":37563,"tokens_out":3755,"would_cite":true,"duration_ms":43488,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G10","62G20","62G35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["differential privacy","scale testing","Siegel–Tukey test","rank transformations","percentile modification","nonparametric hypothesis testing","Wald–Wolfowitz theorem","signed-rank test"],"falsifier":"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 $Q \\in \\{0,\\dots,n-1\\}$.","tokens_in":36431,"feed_emoji":"🔒","tokens_out":12659,"duration_ms":146706,"temperature":0.7,"pith_summary":"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.","feed_headline":"Privatized rank tests compare spreads without leaking group sizes","feed_subtitle":"Laplace noise on Siegel–Tukey ranks keeps type I error in check while p-values come from the normal table.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original center-outward rank-sum procedure for comparing spread on which the RPST ranking is built.","marker":"[20]"},{"why":"Introduces percentile modification, the truncation of central ranks that the RPST tests adopt.","marker":"[14]"},{"why":"Provides the private rank-test framework and the conservative private estimate of the smaller group size that controls type I error here.","marker":"[8]"},{"why":"Defines the private signed-rank test that Section 5 extends with rank transformations and percentile modification.","marker":"[7]"},{"why":"Gives the Laplace mechanism and the sensitivity-calibration principle used to privatize the rank sum.","marker":"[11]"},{"why":"Supplies the basic composition theorem used to combine the noise on the statistic with the noise on the group-size estimate.","marker":"[10]"},{"why":"Provides the permutation central limit theorem (Theorem C.2) used to prove asymptotic normality of the non-private rank sum.","marker":"[23]"},{"why":"Formulates the Lyapunov central limit theorem used for the signed-rank extension and moment conditions.","marker":"[18]"},{"why":"Supplies the general-purpose differentially private testing baseline used in the power comparisons.","marker":"[15]"}],"fun_headline_variants":["Laplace noise on Siegel–Tukey ranks yields private spread tests","Private scale tests via percentile-modified ranks and rank noise","Differentially private scale tests control error via rank noise","Private Siegel–Tukey scale tests survive Laplace noise addition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Laplace noise on Siegel–Tukey ranks yields private spread tests","Private scale tests via percentile-modified ranks and rank noise","Differentially private scale tests control error via rank noise","Private Siegel–Tukey scale tests survive Laplace noise addition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001825,"raw_usage":{"total_tokens":7134,"prompt_tokens":856,"completion_tokens":6278,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":6208}},"tokens_in":472,"tokens_out":6278,"duration_ms":47482,"temperature":1.0,"reasoning_tokens":6208,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:07:15.934756+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $Q \\in \\{0,\\dots,n-1\\}$.","supporting_citations":[{"cited_title":"and Tukey, J","cited_arxiv_id":null,"evidence_quote":"Supplies the original center-outward rank-sum procedure for comparing spread on which the RPST ranking is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces percentile modification, the truncation of central ranks that the RPST tests adopt."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the private rank-test framework and the conservative private estimate of the smaller group size that controls type I error here."},{"cited_title":"A Differentially Private Wilcoxon Signed-Rank Test","cited_arxiv_id":"1809.01635","evidence_quote":"Defines the private signed-rank test that Section 5 extends with rank transformations and percentile modification."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Laplace mechanism and the sensitivity-calibration principle used to privatize the rank sum."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the basic composition theorem used to combine the noise on the statistic with the noise on the group-size estimate."},{"cited_title":"and Wolfowitz, J","cited_arxiv_id":null,"evidence_quote":"Provides the permutation central limit theorem (Theorem C.2) used to prove asymptotic normality of the non-private rank sum."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Formulates the Lyapunov central limit theorem used for the signed-rank extension and moment conditions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the general-purpose differentially private testing baseline used in the power comparisons."}],"review_version":1}