  Testing large covariance matrices is of fundamental importance in statistical
analysis with high-dimensional data. In the past decade, three types of test
statistics have been studied in the literature: quadratic form statistics,
maximum form statistics, and their weighted combination. It is known that
quadratic form statistics would suffer from low power against sparse
alternatives and maximum form statistics would suffer from low power against
dense alternatives. The weighted combination methods were introduced to enhance
the power of quadratic form statistics or maximum form statistics when the
weights are appropriately chosen. In this paper, we provide a new perspective
to exploit the full potential of quadratic form statistics and maximum form
statistics for testing high-dimensional covariance matrices. We propose a
scale-invariant power enhancement test based on Fisher's method to combine the
p-values of quadratic form statistics and maximum form statistics. After
carefully studying the asymptotic joint distribution of quadratic form
statistics and maximum form statistics, we prove that the proposed combination
method retains the correct asymptotic size and boosts the power against more
general alternatives. Moreover, we demonstrate the finite-sample performance in
simulation studies and a real application.
