Empirical Likelihood Test for Diagonal Symmetry

Xin Dang; Yongli Sang

REVIEW 3 major objections 6 minor 39 references

Empirical Likelihood Test for Diagonal Symmetry

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A useful JEL extension for diagonal symmetry with a genuine proof gap at the root-existence lemma; deserves refereeing but not acceptance as-is. read the letter →

arxiv 1908.06892 v1 pith:AFXIFWPO submitted 2019-08-19 stat.ME

Yongli Sang , Xin Dang This is my paper · ORCID

classification stat.ME MSC 62G3562G20

keywords diagonalsymmetrycentralenergydistancejackknifeempiricallikelihoodU-statisticsWilkstheoremnonparametrictestchi-squarecalibration

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 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.

What carries the argument

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 $1$.

What would settle it

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.

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Extended reading notes

Core claim

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 $ heta$ through empirical likelihood constraints, expanding the estimating equations, and showing after algebra that the limiting distribution is a single chi-square degree of freedom.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

The 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.

The 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.

Who 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.

Referee Report

3 major / 6 minor

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.

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 (3)

[Appendix, Lemma 5.3 and Eq. (5)] 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.
[Appendix, proof of Theorem 2.2] 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 3, Table 2 and accompanying text] 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.

minor comments (6)

[Introduction, paragraph 2] The text says 'The reminder of the paper is organized as follows'; 'reminder' should be 'remainder'.
[Abstract] The phrase 'with degree freedom of one' should be 'with one degree of freedom'.
[Table 2 header] The header says 'Simension'; this should be 'Dimension'.
[Appendix, Lemma 5.2] 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}.
[Table 1] 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.
[Proof of Theorem 2.2] The notation in the displayed sum uses S̃_i in one place and S̃_k in another; the subscripts should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the chi-square calibration rests on external JEL and root-existence lemmas, not on fitted inputs or self-citations.

full rationale

The derivation chain is not circular. The null hypothesis H0: X D= -X is reduced to E||X+X'|| = E||X-X'|| using the external energy-distance characterization of Székely and Móri; this is an independent mathematical result, not an assumption of the paper. The JEL statistic is built from jackknife pseudo-values whose sample means are exactly U1 and U2, and the equality EU1 = EU2 is algebraically equivalent to the constraint tested; this is a standard construction, not a fitted parameter renamed as a prediction. Theorem 2.1's chi-square calibration is obtained by importing two external asymptotic results: the JEL expansion of Jing, Yuan and Zhou (2009) (reference [15]) and a root-existence lemma of Liu, Liu and Zhou (2018) (Lemma 5.3, reference [18]). Neither lemma is stated to depend on the present paper's data or fitted values, and neither is authored by the present authors. The only self-citations ([28], [29]) appear in the introduction as examples of JEL applications and are not load-bearing in the proof. The remaining algebra (matrix W, eigenvalues 0 and 1) is a direct calculation from the asymptotic covariance and does not import the conclusion. Whether Lemma 5.3's conditions are fully verified is a correctness/rigor concern, not a circularity; no equation in the paper reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The test has no fitted constants. It stands on the energy-distance characterization of diagonal symmetry, the U-statistic CLT, the jackknife empirical likelihood Wilks theorem, and the split-balance condition C2. The most fragile imported ingredient is Lemma 5.3 of Liu, Liu and Zhou (2018), which supplies the root expansion for the combined estimating equations.

assumptions (5)

domain assumption E||X+X'|| = E||X-X'|| holds if and only if X has the same distribution as -X.
Used in Section 1 to convert the symmetry hypothesis H0 into a moment equality; cited to Szekely and Mori (2001), not proved in the paper.
domain assumption Conditions C1: 0 < sigma_gk < infinity for k=1,2.
Assumed before Theorem 2.1; ensures the U-statistic CLT and nondegenerate jackknife pseudo-values.
domain assumption Condition C2: n_k/n -> alpha_k > 0 with alpha_1 + alpha_2 = 1.
Balanced split is needed so both U-statistics contribute and the asymptotic matrix W is nonsingular.
standard math JEL asymptotic expansions from Jing, Yuan and Zhou (2009) and root-existence lemma from Liu, Liu and Zhou (2018).
Lemma 5.2 and Lemma 5.3 are imported and used without full proof in the appendix.
standard math Hoeffding's U-statistic central limit theorem.
Lemma 5.1 is credited to Hoeffding (1948), but no corresponding entry appears in the reference list.

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Cite this review

Pith. "Pith review of Empirical Likelihood Test for Diagonal Symmetry." pith.science (2026). https://pith.science/paper/AFXIFWPO

@misc{pith2026190806892,
  author       = {Pith},
  title        = {Pith review of: Empirical Likelihood Test for Diagonal Symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AFXIFWPO}},
  note         = {Machine review of arXiv:1908.06892}
}

abstract

Energy distance is a statistical distance between the distributions of random variables, which characterizes the equality of the distributions. Utilizing the energy distance, we develop a nonparametric test for the diagonal symmetry, which is consistent against any fixed alternatives. The test statistic developed in this paper is based on the difference of two $U$-statistics. By applying the jackknife empirical likelihood approach, the standard limiting chi-square distribution with degree freedom of one is established and is used to determine critical value and $p$-value of the test. Simulation studies show that our method is competitive in terms of empirical sizes and empirical powers.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Antille, A., Kersting, G., and Zucchini, W. (1982). Test s symmetry. J. Amer. Statist. Assoc. , 77, 639-646

work page 1982

[2] [2]

Kersting, G., and Wrigh t, A

Bhattacharya, P., Gastwirth, J. Kersting, G., and Wrigh t, A. (1982). Two modiﬁed Wilcoxon test for symmetry about an unknown location parameter. Biometrika, 69, 377-382. 11

work page 1982

[3] [3]

Boos, D. (1982). A test for symmetry associated with the H odges-Lehmann estimator. J. Amer. Statist. Assoc. , 77, 647-651

work page 1982

[4] [4]

and Masaro, J

Cabilio, P. and Masaro, J. (1996). A simple test of symmet ry about an unknown median. Canad. J. Statis. , 24(3), 349-361

work page 1996

[5] [5]

and Van Keilegom, I

Cao, R. and Van Keilegom, I. (2006). Empirical likelihoo d tests for two-sample problems via nonparametric density estimation. Can. J. Stat. , 34(1), 61-77

work page 2006

[6] [6]

and Zhu, L

Chen, F., Meintanis, S.G. and Zhu, L. (2019). On some char acterizations and multidimensional criteria for testing homogeneity, symmetry and independen ce. J. Multivariate Anal. , 173, 125- 144

work page 2019

[7] [7]

Chen, J., Variyath, A. M. and Abraham, B. (2008). Adjuste d empirical likelihood and its properties. J. Comput. Graph. Statist. 17 (2), 426-443

work page 2008

[8] [8]

Doksum, K., Fenstad, G., and Aaberge, R. (1977). Plots an d tests for symmetry. Biometrika, 64(3), 473-487

work page 1977