The k-sample Behrens-Fisher problem for high-dimensional data with model free assumption

The problem of testing the equality of k-sample mean vectors with different covariance matrices, known as the Behrens-Fisher (BF) problem for k-sample, is a significant issue in statistics. Hu and Bai (2017) proposed a test statistic that operates under a factor-like model structure assumption and demonstrated its normal limit. Building on this work, we further explore the asymptotic properties of the test statistic. We prove that the asymptotic null distribution of the test statistic is a Chi-square-type mixture distribution under a model-free assumption and establish its asymptotic power under a full alternative hypothesis. Moreover, we show that the asymptotic null distribution of the test statistic is either normal or a weighted sum of normal and Chi-square random variables, depending on the convergence rate of the eigenvalues of the covariance matrix with model free assumption. To address practical challenges in high-dimensional data, we propose a new weighted bootstrap procedure that is simple to implement. Simulation studies demonstrate that our proposed test procedure outperforms existing methods in terms of size control under various settings. Furthermore, real data applications illustrate the applicability of our test procedure to a variety of high-dimensional data analysis problems.