  We investigate one/two-sample mean tests for high-dimensional compositional
data when the number of variables is comparable with the sample size, as
commonly encountered in microbiome research. Existing methods mainly focus on
max-type test statistics which are suitable for detecting sparse signals.
However, in this paper, we introduce a novel approach using sum-type test
statistics which are capable of detecting weak but dense signals. By
establishing the asymptotic independence between the max-type and sum-type test
statistics, we further propose a combined max-sum type test to cover both
cases. We derived the asymptotic null distributions and power functions for
these test statistics. Simulation studies demonstrate the superiority of our
max-sum type test statistics which exhibit robust performance regardless of
data sparsity.
