  We consider the five classes of multivariate statistical problems identified
by James (1964), which together cover much of classical multivariate analysis,
plus a simpler limiting case, symmetric matrix denoising. Each of James'
problems involves the eigenvalues of $E^{-1}H$ where $H$ and $E$ are
proportional to high dimensional Wishart matrices. Under the null hypothesis,
both Wisharts are central with identity covariance. Under the alternative, the
non-centrality or the covariance parameter of $H$ has a single eigenvalue, a
spike, that stands alone. When the spike is smaller than a case-specific phase
transition threshold, none of the sample eigenvalues separate from the bulk,
making the testing problem challenging. Using a unified strategy for the six
cases, we show that the log likelihood ratio processes parameterized by the
value of the sub-critical spike converge to Gaussian processes with logarithmic
correlation. We then derive asymptotic power envelopes for tests for the
presence of a spike.
