  Tests based on sample mean vectors and sample spatial signs have been studied
in the recent literature for high dimensional data with the dimension larger
than the sample size. For suitable sequences of alternatives, we show that the
powers of the mean based tests and the tests based on spatial signs and ranks
tend to be same as the data dimension grows to infinity for any sample size,
when the coordinate variables satisfy appropriate mixing conditions. Further,
their limiting powers do not depend on the heaviness of the tails of the
distributions. This is in striking contrast to the asymptotic results obtained
in the classical multivariate setup. On the other hand, we show that in the
presence of stronger dependence among the coordinate variables, the spatial
sign and rank based tests for high dimensional data can be asymptotically more
powerful than the mean based tests if in addition to the data dimension, the
sample size also grows to infinity. The sizes of some mean based tests for high
dimensional data studied in the recent literature are observed to be
significantly different from their nominal levels. This is due to the
inadequacy of the asymptotic approximations used for the distributions of those
test statistics. However, our asymptotic approximations for the tests based on
spatial signs and ranks are observed to work well when the tests are applied on
a variety of simulated and real datasets.
