We consider the problem of testing the null hypothesis of sphericity for a high-dimensional covariance matrix against the alternative of a finite (unspecified) number of symmetry-breaking directions (multispiked alternatives) from the point of view of the asymptotic theory of statistical experiments. The region lying below the so-called phase transition or impossibility threshold is shown to be a contiguity region. Simple analytical expressions are derived for the asymptotic power envelope and the asymptotic powers of existing tests. These asymptotic powers are shown to lie very substantially below the power envelope; some of them even trivially coincide with the size of the test. In contrast, the asymptotic power of the likelihood ratio test is shown to be uniformly close to the same.
Based on joint work with Marcelo Moreira, Fundação Getulio Vargas, Rio de Janeiro, and Alexei Onatski, University of Cambridge. Onatski, A., Moreira, M. and Hallin, M. (2013). Asymptotic power of sphericity tests for high-dimensional data. Annals of Statistics 41, 1204-1231. Onatski, A., Moreira, M. and Hallin, M. (2013). Signal detection in high dimension: the multispiked case. arXiv:1210.5663.