A <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$U$$</EquationSource> </InlineEquation>-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, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation>, exceeds the sample sizes, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$n_i,\; i=1, 2$$</EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$n_i, p \rightarrow \infty $$</EquationSource> </InlineEquation>. However, the validity of the test statistic when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation> is fixed but large, including <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$p > n_i$$</EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$U$$</EquationSource> </InlineEquation>-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. Copyright The Institute of Statistical Mathematics, Tokyo 2014