An efficient and distribution-free symmetry test for high-dimensional data based on energy statistics and random projections

Testing the departures from symmetry is a critical issue in statistics. Over the last two decades, substantial effort has been invested in developing tests for central symmetry in multivariate and high-dimensional contexts. Traditional tests, which rely on Euclidean distance, face significant challenges in high-dimensional data. These tests struggle to capture overall central symmetry and are often limited to verifying whether the distribution's center aligns with the coordinate origin, a problem exacerbated by the “curse of dimensionality.” Furthermore, they tend to be computationally intensive, often making them impractical for large datasets. To overcome these limitations, we propose a nonparametric test based on the random projected energy distance, extending the energy distance test through random projections. This method effectively reduces data dimensions by projecting high-dimensional data onto lower-dimensional spaces, with the randomness ensuring maximum preservation of information. Theoretically, as the number of random projections approaches infinity, the risk of power loss from inadequate directions is mitigated. Leveraging U-statistic theory, our test's asymptotic null distribution is standard normal, which holds true regardless of the data dimensionality relative to sample size, thus eliminating the need for re-sampling to determine critical values. For computational efficiency with large datasets, we adopt a divide-and-average strategy, partitioning the data into smaller blocks of size m. Within each block, the estimates of the random projected energy distance are normally distributed. By averaging these estimates across all blocks, we derive a test statistic that is asymptotically standard normal. This method significantly reduces computational complexity from quadratic to linear in sample size, enhancing the feasibility of our test for extensive data analysis. Through extensive numerical studies, we have demonstrated the robust empirical performance of our test in terms of size and power, affirming its practical utility in statistical applications for high-dimensional data.