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Jackknife Empirical Likelihood Approach for K-sample Tests

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Pith's one-line read A K-sample jackknife empirical likelihood test based on categorical Gini correlation is derived, with a chi-square_{K-1} null limit that avoids permutation.

arxiv 1908.00477 v1 pith:ZPQOXVCM submitted 2019-08-01 stat.ME

Yongli Sang , Xin Dang , Yichuan Zhao This is my paper · ORCID
classification stat.ME
keywords categoricalapproachcorrelationempiricalginijackknifelikelihoodmethod
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A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tackles a classic question: do K groups of numbers or vectors come from the same distribution? Existing methods like the energy distance need a permutation procedure to get critical values, which is slow. The authors recast the problem as checking whether a numerical variable (the data) is independent of a categorical variable (the group label). They use a measure called categorical Gini correlation, which is zero exactly when the groups are identically distributed.

To test this, they estimate the relevant distances with U-statistics and then apply jackknife empirical likelihood (JEL). JEL is a nonparametric technique that turns a complicated test statistic into a likelihood ratio based on leave-one-out pseudo-values. The authors show that, under the null hypothesis, minus twice the log of this ratio converges to a chi-square distribution with K-1 degrees of freedom. This gives a simple way to compute p-values without permutation.

Simulations compare the new test, called JEL-S, with five existing tests across normal, heavy-tailed, and skewed distributions, in dimensions 1, 3, and 6. For scale differences, JEL-S often has the highest power. For location differences, it is weaker than tests based on distribution functions. In heavy-tailed and skewed cases, its Type I error can run above the nominal level at sample size 50, a point the authors acknowledge. A real data example on banknote authentication illustrates the method.

Extended reading notes

Core claim

Theorem 2.1: Under H0 and conditions C1 and C2, -2 log R converges in distribution to chi-square with K-1 degrees of freedom. If correct, the proposed test statistic has a standard chi-square null limit and can be used to compute p-values without permutation.

Load-bearing premise

Condition C1 (0 < sigma_{g_k} < infinity) assumes finite variance of the distance-kernel projections for every group; the U-statistic CLT in Lemma 6.1 and the JEL expansion both rely on it. When data are heavy-tailed enough to violate or strain this condition, the chi-square approximation can be poor, consistent with the reported Type I error inflation (up to 0.089 at nominal 0.05) for t5 and exponential distributions at n=50.

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

The test statistic itself has no fitted free parameters. The proof imports several standard results (Hoeffding CLT, JEL expansions, Shi's asymptotic independence, and a root-existence lemma from Liu et al.) and assumes finite variance conditions C1-C2. The main unproved algebraic step is the matrix identity in the appendix that fixes the degrees of freedom.

assumptions (5)
  • domain assumption Condition C1: 0 < sigma_{g_k} < infinity for all k
    Assumed in Theorem 2.1 and Theorem 2.2 to ensure the CLT for U-statistics and the consistency of the test.
  • domain assumption Condition C2: n_k/n -> alpha_k > 0 and sum alpha_k = 1
    Balanced asymptotic sample fractions are needed for the joint CLT and for matrix inversions in the proof.
  • standard math Existence of a root theta near theta0 for the K+2 estimating equations (Lemma 6.3), imported from Liu, Liu and Zhou (2018)
    Used without proof in the JEL proof; standard for jackknife empirical likelihood with U-statistics.
  • standard math Algebraic identities A^T W0 A = A and eigenvalues of Sigma0 A are {0,0,1,...,1}
    Stated in the proof of Theorem 2.1 without derivation; these identities set the degrees of freedom K-1.
  • standard math Asymptotic independence of jackknife pseudo-values (Shi 1984)
    Used to justify the joint CLT of W(eta0) in the proof of Theorem 2.1.

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Pith. "Pith review of Jackknife Empirical Likelihood Approach for K-sample Tests." pith.science (2026). https://pith.science/paper/ZPQOXVCM

@misc{pith2026190800477,
  author       = {Pith},
  title        = {Pith review of: Jackknife Empirical Likelihood Approach for K-sample Tests},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZPQOXVCM}},
  note         = {Machine review of arXiv:1908.00477}
}
abstract

The categorical Gini correlation is an alternative measure of dependence between a categorical and numerical variables, which characterizes the independence of the variables. A nonparametric test for the equality of K distributions has been developed based on the categorical Gini correlation. By applying the jackknife empirical likelihood approach, the standard limiting chi-square distribution with degree freedom of $K-1$ is established and is used to determine critical value and $p$-value of the test. Simulation studies show that the proposed method is competitive to existing methods in terms of power of the tests in most cases. The proposed method is illustrated in an application on a real data set.

Figures

Figures reproduced from arXiv: 1908.00477 by the authors.

Figure 1
Figure 1. Density of each of variables. From [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Empirical Likelihood Test for Diagonal Symmetry

    stat.ME 2019-08 conditional novelty 6.0 of 10

    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.

Reference graph

Works this paper leans on

42 extracted references · 38 canonical work pages · cited by 1 Pith paper

  1. [1]

    W., and Darling, D

    Anderson, T. W., and Darling, D. A. (1952). Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes. Ann. Math. Statis. 23, 193-212. 17

    work page 1952

  2. [2]

    and Titterington, D.M

    Anderson, N.H., Hall, P. and Titterington, D.M. (1994). Two-sample test statistics for mea- suring discrepancies between two multivariate probability density functions using kernel-based density estimates. J. Multivariate Anal. 50, 41-54

    work page 1994

  3. [3]

    and Kaufman, S

    Brill, B. and Kaufman, S. based in part on an earlier implementation by Ruth Heller and Yair Heller. (2017). HHG: Heller-Heller-Gorfine Tests of Independence and Equality of Distribu- tions. R package version 2.2. https://CRAN.R-project.org/package=HHG

    work page 2017

  4. [4]

    and Van Keilegom, I

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

    work page 2006

  5. [5]

    and Zhou, W

    Chen, B.B., Pan, G.M., Yang, Q. and Zhou, W. (2015). Large dimensional empirical likelihood. Statist. Sinica, 25, 1659-1677

    work page 2015

  6. [6]

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

    work page 2008

  7. [7]

    and Zhou, W

    Cheng, C.H., Liu, Y., Liu, Z. and Zhou, W. (2018). Balanced augmented jackknife empirical likelihood for two sample U-statistics. Sci. China Math. , 61, 1129-1138

    work page 2018

  8. [8]

    A new Gini correlation between quantitative and qualitative variables

    Dang, X., Nguyen, D., Chen, X. and Zhang, J. (2019). A new Gini correlation between quantitative and qualitative variables. arXiv:1809.09793

    work page Pith review arXiv 2019

Show all 42 references
  1. [9]

    Darling, D.A. (1957). The Kolomogorov-Smirnov, Cram´ er-von Mises tests.Ann. Math. Stat. , 28(4), 823-838

    1957

  2. [10]

    and McKeague, I

    Einmahl, J. and McKeague, I. (2003). Empirical likelihood based hypothesis testing. Bernoulli, 9(2), 267-290

    2003

  3. [11]

    and Owen, A

    Emerson S. and Owen, A. (2009). Calibration of the empirical likelihood method for a vector mean. Electron. J. Statist., 3, 1161-1192

    2009

  4. [12]

    and Mu˜ noz Garc` ıa, J

    Fern´ andez, V., Jim` enez Gamerro, M. and Mu˜ noz Garc` ıa, J. (2008). A test for the two-sample problem based on empirical characteristic functions. Comput. Stat. Data An. , 52, 3730-3748

    2008

  5. [13]

    and Gorfine, M

    Heller, R., Heller, Y. and Gorfine, M. (2013). A consistent multivariate test of association based on ranks of distances. Biometrika 100(2), 503-510

    2013

  6. [14]

    and Gorfine, M

    Heller, R., Heller, Y., Kaofman, S., Brill, B. and Gorfine, M. (2016). Consistent distribution- free K-sample and independence test for univariate random variables. J. Mach. Learn. Res. 17(29), 1-54

    2016

  7. [15]

    and Liu, J

    Jiang, B., Ye, C. and Liu, J. (2015). Nonparametric K-sample tests via dynamic slicing. J. Amer. Statist. Assoc. 110, 642-653

    2015

  8. [16]

    and Zhou, W

    Jing, B., Yuan, J. and Zhou, W. (2009). Jackknife empirical likelihood. J. Amer. Statist. Assoc. 104, 1224-1232

    2009

  9. [17]

    Kiefer, J. (1959). k-sample analogues of the Kolmogorov-Simirnov, Cram´er-von Mises tests. Ann. Math. Statist. 30, 420-447

    1959

  10. [18]

    and Wallis, W.A

    Kruskal, W.H. and Wallis, W.A. (1952). Use of ranks in one-criterion analysis of variance. J. Amer. Statist. Assoc. 47, 583-621

    1952

  11. [19]

    Liu, X., Wang, Q., and Liu, Y. (2017). A consistent jackknife empirical likelihood test for distribution functions. Ann. Inst. Statist. Math. 69(2), 249-269. 18

    2017

  12. [20]

    and Zhou, W

    Liu, Y., Liu, Z. and Zhou, W. (2018). A test for equality of two sample distributions via integrating characteristic functions. Statist. Sinica, DOI: 10.5705/ss.202017.0236

    2018

  13. [21]

    and Zhou, W

    Liu, Z., Xia, X. and Zhou, W. (2015). A test for equality of two distributions via jackknife empirical likelihood and characteristic functions. Comput. Statist. Data Anal. 92, 97-114

    2015

  14. [22]

    L., D¨ orksen, H., Hildebrand, R., Gillich, E., Hofmann, J

    Lohweg, V., Hoffmann, J. L., D¨ orksen, H., Hildebrand, R., Gillich, E., Hofmann, J. and Schaede, J. (2013). Banknote Authentication with Mobile Devices. Proc. SPIE 8665, Media Watermarking, Security, and Forensics

    2013

  15. [23]

    and de U˜ na-´Alvarez, J

    Mart` ınez-Camblor, P. and de U˜ na-´Alvarez, J. (2009). Non-parametrick-sample tests: Density functions vs distribution functions. Comput. Statist. Data Anal. 53, 3344-3357

    2009

  16. [24]

    Owen, A. (1988). Empirical likelihood ratio confidence intervals for single functional. Biometrika 75, 237-249

    1988

  17. [25]

    Owen, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18, 90-120

    1990

  18. [26]

    Puri, M. L. and Sen, P. K. (1971). Nonparametric Methods in Multivariate Analysis , John Wiley and Sons, New York

    1971

  19. [27]

    and Jing, B.Y

    Qin, G.S. and Jing, B.Y. (2001). Empirical likelihood for censored linear regression. Scand. J. Statist. 28, 661-673

    2001

  20. [28]

    and Lawless, J

    Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating functions. Ann. Statist. 22, 300-325

    1994

  21. [29]

    and Sz´ekely, G.J

    Rizzo, M.L. and Sz´ekely, G.J. (2010). Disco Analysis: A nonparametric extension of analysis of variance. Ann. Appl. Stat. 4, 1034-1055

    2010

  22. [30]

    and Sz´ekely, G.J

    Rizzo, M. and Sz´ekely, G.J. (2017). E-statistics: multivariate inference via the energy of data. R package version 1.7-2. https://CRAN.R-project.org/package=energy

    2017

  23. [31]

    and Zhao, Y

    Sang, Y., Dang, X. and Zhao, Y. (2019) Jackknife empirical likelihood methods for Gini correlation and their equality testing. J. Statist. Plann. Inference , 199, 45-59

    2019

  24. [32]

    Scholz, F. W. and Stpephens, M.A. (1987). K-sample Anderson-Darling tests. J. Amer. Statist. Assoc. 82(399), 918-924

    1987

  25. [33]

    Scholz, F. W. and Zhu, A. (2018). K-sample rank tests and their combinations. R package version 1.2-8. https://CRAN.R-project.org/package=kSamples

    2018

  26. [34]

    Shi, X. (1984). The approximate independence of jackknife pseudo-values and the bootstrap Methods. J. Wuhan Inst. Hydra-Electric Engineering , 2, 83-90

    1984

  27. [35]

    and Rizzo, M.L

    Sz´ ekely, G.J. and Rizzo, M.L. (2004). Testing for equal distributions in high dimension, In- terStat, Nov. (5)

    2004

  28. [36]

    and Rizzo, M.L

    Sz´ ekely, G.J. and Rizzo, M.L. (2013). Energy statistics: A class of statistics based on distances. J. Stat. Plan. Infer. , 143, 1249-1272

    2013

  29. [37]

    and Rizzo, M.L

    Sz´ ekely, G.J. and Rizzo, M.L. (2017). The energy of data, Ann. Rev. Stat. Appl. , 4 (1), 447-479

    2017

  30. [38]

    and ABB Corporate Research (2018)

    Varadhan R., Johns Hopkins University, Borchers, H.W. and ABB Corporate Research (2018). dfoptim: Derivative-Free Optimization. R package version 2018.2-1. https://CRAN.R- project.org/package=dfoptim. 19

    2018

  31. [39]

    and Deng, M

    Wan,Y., Liu, Z. and Deng, M. (2018). Empirical likelihood test for equality of two distributions using distance of characteristic functions. Statistics, DOI: 10.1080/02331888.2018.1520855

    2018

  32. [40]

    and Jing, B.Y

    Wang, Q.H. and Jing, B.Y. (1999). Empirical likelihood for partial linear models with fixed designs. Statist. Probab. Lett. 41, 425-433

    1999

  33. [41]

    Wood, A.T.A., Do, K.A., Broom, N.M. (1996). Sequential linearization of empirical likelihood constraints with application to U-statistics. J. Comput. Graph. Statist. 5, 365-385

    1996

  34. [42]

    and Wu, Y

    Zhang, J. and Wu, Y. (2007). K-sample tests based on the likelihood ratio. Comput. Statist. Data Anal. 51, 4682-4691. 20

    2007

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