  To assess whether there is some signal in a big database, aggregate tests for
the global null hypothesis of no effect are routinely applied in practice
before more specialized analysis is carried out. Although a plethora of
aggregate tests is available, each test has its strengths but also its blind
spots. In a Gaussian sequence model, we study whether it is possible to obtain
a test with substantially better consistency properties than the likelihood
ratio (i.e., Euclidean norm based) test. We establish an impossibility result,
showing that in the high-dimensional framework we consider, the set of
alternatives for which a test may improve upon the likelihood ratio test --
that is, its superconsistency points -- is always asymptotically negligible in
a relative volume sense.
