  This paper establishes the asymptotic independence between the quadratic form
and maximum of a sequence of independent random variables. Based on this
theoretical result, we find the asymptotic joint distribution for the quadratic
form and maximum, which can be applied into the high-dimensional testing
problems. By combining the sum-type test and the max-type test, we propose the
Fisher's combination tests for the one-sample mean test and two-sample mean
test. Under this novel general framework, several strong assumptions in
existing literature have been relaxed. Monte Carlo simulation has been done
which shows that our proposed tests are strongly robust to both sparse and
dense data.
