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Test for diagonal symmetry in high dimension

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  • Sang, Yongli

Abstract

Utilizing the energy distance and energy statistics, Sang and Dang (2020) proposed a test statistic as a difference of two U-statistics for the diagonal symmetry test of a p-vector X. Under the regular setting where the dimensionality of the random vector is fixed, the test statistic is a degenerate U-statistic and hence converges to a mixture of chi-squared distributions. In this paper, we test the diagonal symmetry of X in a more realistic setting where both the sample size and the dimensionality are diverging to infinity. Our theoretical results reveal that the degenerate U-statistic admits a central limit theorem in the high dimensional setting and the accuracy of normal approximation can increase with dimensionality. We then construct a powerful and consistent test for the diagonal symmetry problem based on the asymptotic normality. Simulation studies are conducted to illustrate the performances of the test.

Suggested Citation

  • Sang, Yongli, 2024. "Test for diagonal symmetry in high dimension," Statistics & Probability Letters, Elsevier, vol. 205(C).
    • Handle: RePEc:eee:stapro:v:205:y:2024:i:c:s0167715223001840
    DOI: 10.1016/j.spl.2023.109960
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    References listed on IDEAS

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    1. Sang, Yongli & Dang, Xin, 2020. "Empirical likelihood test for diagonal symmetry," Statistics & Probability Letters, Elsevier, vol. 156(C).
    2. Chen, Feifei & Meintanis, Simos G. & Zhu, Lixing, 2019. "On some characterizations and multidimensional criteria for testing homogeneity, symmetry and independence," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 125-144.
    3. Henderson, Daniel J. & Parmeter, Christopher F., 2015. "A consistent bootstrap procedure for nonparametric symmetry tests," Economics Letters, Elsevier, vol. 131(C), pages 78-82.
    4. Lyubchich, Vyacheslav & Wang, Xingyu & Heyes, Andrew & Gel, Yulia R., 2016. "A distribution-free m-out-of-n bootstrap approach to testing symmetry about an unknown median," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 1-9.
    5. Henze, N. & Klar, B. & Meintanis, S. G., 2003. "Invariant tests for symmetry about an unspecified point based on the empirical characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 275-297, November.
    6. Fang, Ying & Li, Qi & Wu, Ximing & Zhang, Daiqiang, 2015. "A data-driven smooth test of symmetry," Journal of Econometrics, Elsevier, vol. 188(2), pages 490-501.
    7. Dai, Xinjie & Niu, Cuizhen & Guo, Xu, 2018. "Testing for central symmetry and inference of the unknown center," Computational Statistics & Data Analysis, Elsevier, vol. 127(C), pages 15-31.
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    Cited by:

    1. Chen, Bo & Chen, Feifei & Wang, Junxin & Qiu, Tao, 2025. "An efficient and distribution-free symmetry test for high-dimensional data based on energy statistics and random projections," Computational Statistics & Data Analysis, Elsevier, vol. 206(C).
    2. Yong Wang & Reza Modarres, 2026. "Testing high dimensional diagonal symmetry," Statistical Papers, Springer, vol. 67(1), pages 1-33, February.

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