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<p>In this paper, we propose a robust nonparametric test for detecting conditional symmetry in the conditional distribution of high-dimensional random vectors. The test statistic employs the weighted Euclidean distance between paired empirical characteristic functions, which is generalized through the random selection of subspaces. This approach addresses the limitations of distance-based tests in high dimensions, which often fail to capture overall conditional symmetry and are typically confined to verifying the alignment of the distribution’s centroid with the origin. As the number of random subspaces increases, our method mitigates information loss resulting from suboptimal subspace selection. Theoretically, if the number of random subspaces tends to infinity, we can obtain the final statistic, thereby avoiding potential power loss caused by suboptimal subspaces. Therefore, the proposed test is consistent with general alternatives. Utilizing <i>U</i>-statistical theory, the asymptotic null distribution of our proposed test is standard normal under the null hypothesis of conditional symmetry, regardless of the parent distributions of the random samples or the relationship between data dimensions and sample sizes. This eliminates the need for resampling procedures to determine critical values. Additionally, the proposed test does not require any moment conditions, enabling it to handle heavy-tailed or outlier data. Simulations demonstrate that our test is robust to the dimensionality of the conditional vector and maintains high efficiency.</p>

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A robust nonparametric test for conditional symmetry in high dimension

  • 期刊论文

  • 发表日期：

    2025年12月5日

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  • Tao Qiu,
  • Jianming Wei,
  • Feifei Chen,
  • Jiakun Jiang

摘要

In this paper, we propose a robust nonparametric test for detecting conditional symmetry in the conditional distribution of high-dimensional random vectors. The test statistic employs the weighted Euclidean distance between paired empirical characteristic functions, which is generalized through the random selection of subspaces. This approach addresses the limitations of distance-based tests in high dimensions, which often fail to capture overall conditional symmetry and are typically confined to verifying the alignment of the distribution’s centroid with the origin. As the number of random subspaces increases, our method mitigates information loss resulting from suboptimal subspace selection. Theoretically, if the number of random subspaces tends to infinity, we can obtain the final statistic, thereby avoiding potential power loss caused by suboptimal subspaces. Therefore, the proposed test is consistent with general alternatives. Utilizing U-statistical theory, the asymptotic null distribution of our proposed test is standard normal under the null hypothesis of conditional symmetry, regardless of the parent distributions of the random samples or the relationship between data dimensions and sample sizes. This eliminates the need for resampling procedures to determine critical values. Additionally, the proposed test does not require any moment conditions, enabling it to handle heavy-tailed or outlier data. Simulations demonstrate that our test is robust to the dimensionality of the conditional vector and maintains high efficiency.