{"id":"ca69fa5b-e44f-4f4f-afc4-5c309f58ba6b","arxiv_id":"1908.06892","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A jackknife empirical likelihood test based on energy distance detects diagonal symmetry with an asymptotic chi-square distribution with one degree of freedom and is consistent against all fixed alternatives.","lead":"This paper proposes a nonparametric test for whether a multivariate distribution is symmetric around zero, using energy distance and a jackknife empirical likelihood calibration. The test's p-values come from a standard chi-square curve, so no permutation resampling is needed.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1's chi-square calibration rests on an imported root-existence lemma that is neither proved nor fully restated; the missing n^{-1/2} rate for (θ̃, λ̃) leaves the remainder term Rjn in (5) unverified.","rationale":"The reader identified the imported Lemma 5.3 and the JEL expansion as the weakest assumption, and my reading agrees. I checked the rest of the proof where I could: with the Jacobian W defined as the negative of the derivative of the estimating equations, the linearized solution for θ̃−θ0 matches the stated expression, and the eigenvalue computation for the quadratic form in (6) is algebraically correct, yielding eigenvalues 0 and 1 under H0. Thus the main unresolved point is not the covariance algebra but the existence and rate of the root η̃ and the size of the remainder Rjn. The paper's assertion that Rjn=oP(n^{-1/2}) is not proved, and the quoted lemma is not sufficient as stated because it only controls θ̃ at an n^{-1/3} rate and says nothing about λ̃. This is a genuine gap in the central proof, though it is plausibly fillable by standard arguments. I am not changing the reader's conditional verdict because the concern does not establish a known false conclusion; it identifies a missing proof step that must be supplied before the theorem is relied upon. The simulation overstatement and the lack of code are secondary reporting concerns and do not affect the asymptotic claim as directly as the missing root-rate argument.","tokens_in":10955,"tokens_out":14269,"duration_ms":146711,"concrete_test":"Independently re-derive the proof of Theorem 2.1 without invoking Lemma 5.3: solve the three estimating equations (4) by a Newton/contractive-mapping argument around η0=(θ0,0,0), using the nonsingular Jacobian W computed in the Appendix and Wjn(η0)=O_P(n^{-1/2}), and prove ||η̃−η0||=O_P(n^{-1/2}) and hence Rjn=oP(n^{-1/2}). If this derivation cannot be completed, the chi-square calibration in Theorem 2.1 is not established; if it can be completed, the imported-lemma concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the chi-square calibration in Theorem 2.1. The proof's bottleneck is Appendix Lemma 5.3, imported from Liu, Liu and Zhou (2018), which asserts existence of a root of the three estimating equations (4). As stated, the lemma only guarantees |θ̃−θ0| < n^{-1/3}; it does not state the rate of λ̃1, λ̃2, or of the full vector η̃−η0. The subsequent expansion (5) requires Rjn = oP(n^{-1/2}), and the text says 'one can easily prove' this without a derivation. If the root rate is only n^{-1/3}, a quadratic remainder is typically O_P(n^{-2/3}), which is not oP(n^{-1/2}); obtaining the needed n^{-1/2} rate requires a separate argument using nonsingularity of the Jacobian W and Wjn(η0)=O_P(n^{-1/2}), and that argument is not supplied. The final quadratic form (6) and the eigenvalue calculation that yields the chi-square limit depend directly on this expansion, so a failure of the root-existence or remainder step would break Theorem 2.1. This is the load-bearing unverified step in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a jackknife empirical likelihood (JEL) test for the hypothesis of diagonal symmetry, H0: X^D = -X, for a d-variate random vector. It uses the known characterization that this hypothesis is equivalent to E||X+X'|| = E||X-X'||, estimates the two expectations by U-statistics computed on two independent subsamples, forms jackknife pseudo-values, and constructs an empirical likelihood ratio with a common mean parameter. The main theoretical results are Theorem 2.1, which claims that the jackknife empirical log-likelihood ratio converges in distribution to a chi-square with one degree of freedom under H0, and Theorem 2.2, which claims consistency against any fixed alternative. The paper reports simulations comparing the proposed method with permutation-based energy and characteristic-function tests, as well as with several univariate symmetry tests.","tokens_in":11206,"tokens_out":14198,"duration_ms":141466,"significance":"If the main theorem is correct, the paper provides a useful permutation-free test of central symmetry about a specified center; the chi-square calibration is a practical advantage, and consistency against all fixed alternatives is a strong theoretical guarantee. The construction has no fitted tuning parameters, the null distribution comes from the standard chi-square table, and the energy-distance characterization is properly attributed to prior work. The simulation study covers several dimensions, heavy-tailed distributions, unequal subsample sizes, and alternatives of both location and scatter type, which is a reasonable empirical evaluation. The main weakness is that the proof of the central result imports a root-existence lemma without proof, and the proof of consistency is only sketched; the simulation summary also misstates one entry in Table 2. No code is shipped, but the simulation design is described in enough detail to be reproduced with reasonable effort.","major_comments":[{"comment":"Lemma 5.3 is imported from Liu, Liu and Zhou (2018) without proof or a precise theorem number, and its statement as printed is incomplete: it refers to a root θ̃ of the system Wjn(θ,λ)=0, j=1,2,3, but bounds only |θ̃−θ0| and says nothing about λ̃1 and λ̃2 or the full vector η̃. The expansion in Eq. (5) and the assertion that Rjn=oP(n^{−1/2}) are the load-bearing steps that turn the existence of a root into the linear approximation used for the chi-square limit; the text says only that this is 'easy to prove'. I do not think the n^{−1/3} rate is by itself fatal, since a quadratic remainder from a root with η̃−η0=O_P(n^{−1/3}) would be O_P(n^{−2/3})=o_P(n^{−1/2}); the real problem is that the manuscript does not prove, or even fully state, the existence and rate of the full root (θ̃,λ̃1,λ̃2). Without that, Theorem 2.1 is not established.","section":"Appendix, Lemma 5.3 and Eq. (5)"},{"comment":"The consistency proof concludes that 'at least one of √n_k(θ_k−θ̃)^2/S̃_k' will diverge, but this requires an argument that θ̃ does not converge to both θ1 and θ2 or otherwise track the alternative in a way that leaves the sum bounded. The proof does not state the probability limit of θ̃ under the alternative, nor does it establish that the EL ratio l diverges rather than converging to a finite limit. A rigorous proof of Theorem 2.2 should show, for example, that the constrained empirical likelihood is exponentially small when a common mean is imposed, or at least that θ̃ is stochastically bounded away from one of the two population means.","section":"Appendix, proof of Theorem 2.2"},{"comment":"Table 2 reports an empirical size of 0.180 for JEL at d=6, n1=n2=20 for the t5(0,Σ) distribution, while the text states that JEL has only a 'slight oversize' problem and that 'the empirical sizes of all methods are fairly close to the nominal levels'. A size of 0.180 is not fairly close to 0.05 and is not slight at the 0.05 level; this distortion should be acknowledged and discussed, or the claim should be restricted to the larger sample sizes. This matters because the simulation section is the paper's evidence that the chi-square calibration is usable in finite samples.","section":"Section 3, Table 2 and accompanying text"}],"minor_comments":[{"comment":"The text says 'The reminder of the paper is organized as follows'; 'reminder' should be 'remainder'.","section":"Introduction, paragraph 2"},{"comment":"The phrase 'with degree freedom of one' should be 'with one degree of freedom'.","section":"Abstract"},{"comment":"The header says 'Simension'; this should be 'Dimension'.","section":"Table 2 header"},{"comment":"The summation index is written as ∑_{i=1}^{n}, but the sum is over i=1,...,n_j; this should be corrected to ∑_{i=1}^{n_j}.","section":"Appendix, Lemma 5.2"},{"comment":"For N(0,1), the rows with n1=40, n2=60 and n1=n2=50 are identical; this is likely a transcription error and should be checked.","section":"Table 1"},{"comment":"The notation in the displayed sum uses S̃_i in one place and S̃_k in another; the subscripts should be made consistent.","section":"Proof of Theorem 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for stat.ME and the self-citations to the authors' related JEL work are appropriate. The main risk is the unproved Appendix Lemma 5.3; if the authors can prove it, or state and verify the needed result from Liu, Liu and Zhou (2018) with the required rates, the central theorem is likely repairable. The Table 2 size distortion and the overly brief proof of Theorem 2.2 should also be addressed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper puts a jackknife empirical likelihood (JEL) test on the energy-distance characterization of diagonal symmetry, with a split-sample device to sidestep degeneracy. The chi-square calibration is a real contribution if it holds: it would give a permutation-free symmetry test with a standard limiting distribution. The split-sample construction for this specific U-statistic is not in the cited literature, and extending JEL to a degenerate setting this way is a legitimate methodological step.\n\nWhat the paper does well: the idea is clean, the test is consistent against fixed alternatives (Theorem 2.2's logic is sound), and the simulations cover a reasonable set of distributions and dimensions. The comparison with permutation-based energy tests and characteristic-function tests is appropriate, and the power results in the non-identity covariance cases are interesting.\n\nThe soft spot is the proof of Theorem 2.1. The stress-test note is right: Lemma 5.3 is imported from Liu, Liu and Zhou (2018) and is not proved or even restated with conditions tailored to this setting. As stated, it only gives |θ̃−θ0| < n^{−1/3}, with no rate for λ̃1, λ̃2, or the full vector. The expansion in (5) requires Rjn = oP(n^{−1/2}), and the text says 'one can easily prove' without showing the argument. The quadratic remainder from an n^{−1/3} root is typically O_P(n^{−2/3}), which is not oP(n^{−1/2}); you need a separate step using nonsingularity of the Jacobian and Wjn(η0)=OP(n^{−1/2}) to upgrade the rate. That step is missing. This is load-bearing, because the chi-square limit depends directly on the expansion and the eigenvalue calculation that follows.\n\nThe simulations also overstate the finite-sample story. Table 2 shows empirical size 0.180 for t5 in d=6 at n=20, and Table 1 has sizes around 0.075–0.078 for heavy-tailed univariate cases. Calling that 'fairly close' is generous. The oversize is real but not catastrophic at larger n; still, the summary should not paper over it. No code is provided, so reproducing the simulation requires reimplementation.\n\nWho is this for? People working on nonparametric symmetry tests or on JEL for degenerate U-statistics. It is a subfield contribution, not a paradigm shift. The central idea is worth engaging with, but the proof needs a serious rewrite: either prove the imported lemma's applicability in this setting or give a self-contained derivation of the n^{−1/2} rate. A serious referee should ask for that. I would not cite this in its current form, but I would send it to peer review with the expectation of heavy revision.","headline":"A useful JEL extension for diagonal symmetry with a genuine proof gap at the root-existence lemma; deserves refereeing but not acceptance as-is.","tokens_in":11720,"tokens_out":1825,"would_cite":false,"duration_ms":18869,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G35","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a jackknife empirical likelihood ratio built on two energy-distance U-statistics converges to a chi-square distribution with one degree of freedom under diagonal symmetry, giving a permutation-free consistent test.","keywords":["diagonal symmetry","central symmetry","energy distance","jackknife empirical likelihood","U-statistics","Wilks theorem","nonparametric test","chi-square calibration"],"falsifier":"Generate many samples under the null hypothesis from a diagonal symmetric heavy-tailed distribution, say a $t_5$ distribution in dimension six with $n_1=n_2=50$, compute the jackknife empirical log-likelihood ratio $l$, and compare the empirical 95th percentile of $l$ across replications to 3.841, the chi-square one-degree-of-freedom critical value; a persistent large gap would contradict Theorem 2.1. One could also record how often the numerical solver for equations (4) fails to locate a root, since the imported lemma guarantees existence only with probability tending to one.","tokens_in":10745,"feed_emoji":"📊","tokens_out":4119,"duration_ms":46674,"temperature":0.7,"pith_summary":"The paper develops a nonparametric test for diagonal symmetry, meaning the distribution of a random vector $X$ equals that of $-X$. It shows that this hypothesis is equivalent to equality of the two expected distances $E\\|X+X'\\|$ and $E\\|X-X'\\|$, which can be estimated by two $U$-statistics computed on a random split of the sample. The test statistic is a jackknife empirical log-likelihood ratio, and the paper's main result is a Wilks-type theorem: under the null hypothesis the ratio converges in distribution to a chi-square random variable with one degree of freedom, so critical values and $p$-values come from a standard table and no permutation procedure is needed. The test is also shown to be consistent against any fixed alternative. Simulations indicate size control close to nominal levels and strong power for scatter-type asymmetry, with weaker power for pure location shifts.","feed_headline":"Distance-based symmetry test needs no permutations","feed_subtitle":"A jackknife empirical likelihood ratio follows chi-square with one degree of freedom, so p-values come from a table.","key_machinery":"The central object is the jackknife empirical log-likelihood ratio $l$, built from two groups of jackknife pseudo-values $\\hat V_i^{(1)}$ and $\\hat V_j^{(2)}$ obtained by leave-one-out deletion from the two energy $U$-statistics $U_1$ and $U_2$. The pseudo-values are approximately independent within and across groups, and the empirical likelihood constraints force the weighted means of the two groups to share a common value $\\theta$. This converts the nonlinear problem of comparing two $U$-statistics into a likelihood ratio problem with linear constraints, and the main technical work is showing that the resulting $l$ has the same asymptotic expansion as a squared standardized sum of two independent statistics, whose covariance matrix after transformation has eigenvalues $0$ and $1$.","core_discovery":"The central claim is Theorem 2.1: under the null hypothesis $X \\stackrel{d}{=} -X$ and conditions C1--C2, the jackknife empirical log-likelihood ratio $l$ satisfies $l \\xrightarrow{d} \\chi^2_1$ as the sample size grows, where C1 requires the first-order projection variances of the two $U$-statistics to be finite and positive, and C2 requires both subsample proportions to stay bounded away from zero. This gives a direct chi-square calibration for the test. Theorem 2.2 states that the test is consistent for any fixed alternative, so its power tends to one as the sample size tends to infinity. The proof works by writing the two $U$-statistics as sums of jackknife pseudo-values, imposing a common mean $\theta$ through empirical likelihood constraints, expanding the estimating equations, and showing after algebra that the limiting distribution is a single chi-square degree of freedom.","pith_inferences":["The same split-sample jackknife construction could be adapted to test equality of any two $U$-statistics whose null hypothesis is a common mean, not just the two energy distances used here, potentially yielding a general framework for comparing symmetric functionals.","The simulation result that power is low for location-shift alternatives suggests the empirical likelihood weights concentrate near the origin; a modified kernel or a deliberately chosen distance weighting might improve location sensitivity while preserving the chi-square calibration.","Because the energy distance characterizes distributional equality, the approach could plausibly extend to other symmetry classes, such as spherical or elliptical symmetry, by replacing the Euclidean norm kernel with a kernel adapted to the invariance group of the target symmetry."],"forward_implications":["Users can compute $p$-values for diagonal symmetry directly from the chi-square table with one degree of freedom, avoiding the computational cost of permutation tests.","The test rejects any fixed asymmetric distribution with probability tending to one as the sample size grows, so it is an omnibus test rather than one targeted at a particular alternative.","The method works in arbitrary dimension $d$ with the Euclidean distance, and the simulations suggest the chi-square calibration remains reliable across dimensions and for unequal subsample sizes.","When the center is specified, the test applies to symmetry about that center by first centering the data; the paper notes that an unknown center would require a profiled version.","The split of the sample into two parts does not affect the limiting distribution as long as both parts grow proportionally, giving flexibility in implementation."],"supporting_citations":[{"why":"Supplies the characterization that diagonal symmetry holds if and only if $E\\|X+X'\\|=E\\|X-X'\\|$, which is the equivalence the test statistic is built on.","marker":"[34]"},{"why":"Provides the jackknife empirical likelihood method, the pseudo-value asymptotic independence, and the expansion of the empirical log-likelihood ratio used in the proof of Theorem 2.1.","marker":"[15]"},{"why":"Supplies the imported root-existence lemma for the three estimating equations, the key external ingredient ensuring the expansion of the likelihood equations has the needed remainder rate.","marker":"[18]"},{"why":"Defines the DISCO energy test used as the main permutation-based comparison method in the simulations.","marker":"[26]"},{"why":"Provides the characteristic-function-based symmetry test used as another permutation-based comparison baseline in the simulations.","marker":"[6]"},{"why":"Establishes the energy statistics framework and the distance-based characterization that motivates using energy distance as the measure of distributional discrepancy.","marker":"[35]"}],"fun_headline_variants":["Jackknife likelihood gives symmetry test a chi-square limit","Nonparametric symmetry test rejects via chi-square table","No permutations: symmetry test uses one chi-square degree","Consistent symmetry test with chi-square p-values from jackknife","Empirical likelihood symmetry test: chi-square calibration"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole chi-square calibration rests on a borrowed technical result, quoted without proof, that the three likelihood equations have a well-behaved root close to the true value with a small remainder; if that result does not hold for the distance-based $U$-statistics, the test's $p$-values would not be valid.","fun_headline_variants_meta":{"raw":{"variants":["Jackknife likelihood gives symmetry test a chi-square limit","Nonparametric symmetry test rejects via chi-square table","No permutations: symmetry test uses one chi-square degree","Consistent symmetry test with chi-square p-values from jackknife","Empirical likelihood symmetry test: chi-square calibration"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000591,"raw_usage":{"total_tokens":2712,"prompt_tokens":827,"completion_tokens":1885,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":1807}},"tokens_in":443,"tokens_out":1885,"duration_ms":13199,"temperature":1.0,"reasoning_tokens":1807,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:32:32.947633+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate many samples under the null hypothesis from a diagonal symmetric heavy-tailed distribution, say a $t_5$ distribution in dimension six with $n_1=n_2=50$, compute the jackknife empirical log-likelihood ratio $l$, and compare the empirical 95th percentile of $l$ across replications to 3.841, the chi-square one-degree-of-freedom critical value; a persistent large gap would contradict Theorem 2.1. One could also record how often the numerical solver for equations (4) fails to locate a root, since the imported lemma guarantees existence only with probability tending to one.","supporting_citations":[{"cited_title":"and M´ ori, T.F","cited_arxiv_id":null,"evidence_quote":"Supplies the characterization that diagonal symmetry holds if and only if $E\\|X+X'\\|=E\\|X-X'\\|$, which is the equivalence the test statistic is built on."},{"cited_title":"and Zhou, W","cited_arxiv_id":null,"evidence_quote":"Provides the jackknife empirical likelihood method, the pseudo-value asymptotic independence, and the expansion of the empirical log-likelihood ratio used in the proof of Theorem 2.1."},{"cited_title":"and Zhou, W","cited_arxiv_id":null,"evidence_quote":"Supplies the imported root-existence lemma for the three estimating equations, the key external ingredient ensuring the expansion of the likelihood equations has the needed remainder rate."},{"cited_title":"and Sz´ ekely, G.J","cited_arxiv_id":null,"evidence_quote":"Defines the DISCO energy test used as the main permutation-based comparison method in the simulations."},{"cited_title":"and Zhu, L","cited_arxiv_id":null,"evidence_quote":"Provides the characteristic-function-based symmetry test used as another permutation-based comparison baseline in the simulations."},{"cited_title":"and Rizzo, M.L","cited_arxiv_id":null,"evidence_quote":"Establishes the energy statistics framework and the distance-based characterization that motivates using energy distance as the measure of distributional discrepancy."}],"review_version":1}