  Covariance matrix testing for high dimensional data is a fundamental problem.
A large class of covariance test statistics based on certain averaged spectral
statistics of the sample covariance matrix are known to obey central limit
theorems under the null. However, precise understanding for the power behavior
of the corresponding tests under general alternatives remains largely unknown.
This paper develops a general method for analyzing the power behavior of
covariance test statistics via accurate non-asymptotic power expansions. We
specialize our general method to two prototypical settings of testing identity
and sphericity, and derive sharp power expansion for a number of widely used
tests, including the likelihood ratio tests, Ledoit-Nagao-Wolf's test, Cai-Ma's
test and John's test. The power expansion for each of those tests holds
uniformly over all possible alternatives under mild growth conditions on the
dimension-to-sample ratio. Interestingly, although some of those tests are
previously known to share the same limiting power behavior under spiked
covariance alternatives with a fixed number of spikes, our new power
characterizations indicate that such equivalence fails when many spikes exist.
The proofs of our results combine techniques from Poincar\'e-type inequalities,
random matrices and zonal polynomials.
