  In this paper, we discuss tests for mean vector of high-dimensional data when
the dimension $p$ is a function of sample size $n$. One of the tests, called
the decomposite $T^{2}$-test, in the high-dimensional testing problem is
constructed based on the estimation work of Ledoit and Wolf (2018), which is an
optimal orthogonally equivariant estimator of the inverse of population
covariance matrix under Stein loss function. The asymptotic distribution
function of the test statistic is investigated under a sequence of local
alternatives. The asymptotic relative efficiency is used to see whether a test
is optimal and to perform the power comparisons of tests. An application of the
decomposite $T^{2}$-test is in testing significance for the effect of monthly
unlimited transport policy on public transportation, in which the data are
taken from Taipei Metro System.
