Cross projection test for mean vectors via multiple random splits in high dimensions

The cross projection test (CPT) technique is extended to high-dimensional two-sample mean tests in this article, which was first proposed by Wang and Cui (2024). A data-splitting strategy is required to find the projection directions that reduce the data from high dimensional space to low dimensional space which can well solve the issue of “the curse of dimensionality”. As long as both samples are randomly split once, two correlated cross projection statistics can be established according to the CPT development mechanism, which is similar to all constructed test statistics that exist the correlation caused by multiple random splits. To deal with this issue and improve the performance of empirical powers by eliminating the randomness of data-splitting, we further utilize a powerful Cauchy combination test algorithm based on multiple data-splitting. Theoretically, we prove the asymptotic property of the proposed test statistic. Furthermore, for the sparse alternative case, we apply the power enhancement technique to the ensemble Cauchy combination test-based algorithm in marginal screening for the full data. Numerical studies through Monte Carlo simulations and two real data examples are conducted simultaneously to illustrate the utility of our proposed ensemble algorithm.