  Precision matrix, which is the inverse of covariance matrix, plays an
important role in statistics, as it captures the partial correlation between
variables. Testing the equality of two precision matrices in high dimensional
setting is a very challenging but meaningful problem, especially in the
differential network modelling. To our best knowledge, existing test is only
powerful for sparse alternative patterns where two precision matrices differ in
a small number of elements. In this paper we propose a data-adaptive test which
is powerful against either dense or sparse alternatives. Multiplier bootstrap
approach is utilized to approximate the limiting distribution of the test
statistic. Theoretical properties including asymptotic size and power of the
test are investigated. Simulation study verifies that the data-adaptive test
performs well under various alternative scenarios. The practical usefulness of
the test is illustrated by applying it to a gene expression data set associated
with lung cancer.
