Abstract:

Given multivariate Gaussian data, we are interested in testing a null hypothesis where the covariance matrix is the identity against an alternative hypothesis where the covariance matrix has a few spiked eigenvalues. We consider a setting where the number of spikes grows as a power of p, and where each spiked eigenvalue is only slightly larger than 1. We discover a phase transition: the twodimensional phase space that calibrates the spike sparsity and strengths partitions into the Region of Impossibility and the Region of Possibility. In Region of Impossibility, all tests are (asymptotically) powerless in separating the alternative from the null. In Region of Possibility, there are tests that have (asymptotically) full power. We consider a CuSum test, a tracebased test, an eigenvaluebased Higher Criticism test, and a TracyWidom test, among which the CuSum test and eigenHC test are new. We show that the first two tests have asymptotically full power in Region of Possibility. The study requires careful analysis of the L1distance of our testing problem and delicate Radom Matrix Theory.
