A -statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens-Fisher setting

A two-sample test statistic is presented for testing the equality of mean vectors when the dimension, , exceeds the sample sizes, , and the distributions are not necessarily normal. Under mild assumptions on the traces of the covariance matrices, the statistic is shown to be asymptotically Chi-square distributed when . However, the validity of the test statistic when is fixed but large, including , and when the distributions are multivariate normal, is shown as special cases. This two-sample Chi-square approximation helps us establish the validity of Box's approximation for high-dimensional and non-normal data to a two-sample setup, valid even under Behrens-Fisher setting. The limiting Chi-square distribution of the statistic is obtained using the asymptotic theory of degenerate -statistics, and using a result from classical asymptotic theory, it is further extended to an approximate normal distribution. Both independent and paired-sample cases are considered.