  We consider two alternative tests to the Higher Criticism test of Donoho and
Jin [Ann. Statist. 32 (2004) 962-994] for high-dimensional means under the
sparsity of the nonzero means for sub-Gaussian distributed data with unknown
column-wise dependence. The two alternative test statistics are constructed by
first thresholding $L_1$ and $L_2$ statistics based on the sample means,
respectively, followed by maximizing over a range of thresholding levels to
make the tests adaptive to the unknown signal strength and sparsity. The two
alternative tests can attain the same detection boundary of the Higher
Criticism test in [Ann. Statist. 32 (2004) 962-994] which was established for
uncorrelated Gaussian data. It is demonstrated that the maximal
$L_2$-thresholding test is at least as powerful as the maximal
$L_1$-thresholding test, and both the maximal $L_2$ and $L_1$-thresholding
tests are at least as powerful as the Higher Criticism test.
