Testing in high-dimensional spiked models

We consider five different classes of multivariate statistical problems identified by James (1964). Each of these problems is related to 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 larger than a case-specific phase transition threshold, one of the eigenvalues of $E^{-1}H$ separates from the bulk. This makes the alternative easily detectable, so that reasonable statistical tests have asymptotic power one. In contrast, when the spike is sub-critical, that is lies below the threshold, none of the eigenvalues separates from the bulk, which makes the testing problem more interesting from the statistical perspective. In such 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 use this result to derive the asymptotic power envelopes for tests for the presence of a spike in the data representing each of the five cases in James' classification.