A Behrens–Fisher problem for general factor models in high dimensions

We revisit the well-known Behrens–Fisher problem in an original and challenging high-dimensional framework, and propose a testing procedure which accommodates a low-dimensional latent factor model. The developed inferential framework is general, as it applies to problems where the underlying populations may be non-normal, the dimension of the population mean vectors may highly exceed the sample size, the design may be unbalanced, and the loading factor dimensions may be different. Under a high-dimensional asymptotic regime, combined with fairly weak technical conditions, we show that null limiting distributions of the test statistics follow a weighted mixture of chi-square distributions, which depends only on the spectrum of the noise covariance matrix and the number of latent factors. As these latter are usually unknown in practice, we exploit an estimation procedure which builds on recent advances in random matrix theory. The asymptotic power of the proposed test is established. A numerical study confirms good analytical properties of the new test that compares favorably to existing procedures used in a similar context. Real data applications are demonstrated with a study of a leukemia data set.