A two-sample mean vector projection test for high-dimensional Behrens–Fisher problems

This paper addresses the high-dimensional two-sample Behrens–Fisher problem of testing the equality of mean vectors. Building upon the projection statistic of Huang et al. [Two-sample mean vector projection test in high-dimensional data. Comput Stat. 2024;39:1061–1091], we show that, under relaxed covariance structure conditions, its null limiting distribution is a mixture of chi-square distributions, rather than the normal limit typically assumed in existing approaches. The chi-square-type mixture is then effectively approximated via the Welch–Satterthwaite method, with the involved parameters consistently estimated from the data. Simulation studies demonstrate that the proposed calibration strategy achieves well-controlled Type I error rates and offers competitive power in most settings compared with several existing methods. A real-data application to corneal surface data further illustrates the practical utility of our procedure.