Signal Detection in High Dimension: The Multispiked Case Alexei Onatski, Marcelo J. Moreira and Marc Hallin ∗ October 31, 2018 This paper deals with the local asymptotic structure, in the sense of Le Cam’s asymptotic theory of statistical experiments, of the signal detec- tion problem in high dimension. More precisely, we consider the problem of testing the null hypothesis of sphericity of a high-dimensional covariance matrix against an alternative of (unspecified) multiple symmetry-breaking directions ( multispiked alternatives). Simple analytical expressions for the asymptotic power envelope and the asymptotic powers of previously pro- posed tests are derived. These asymptotic powers are shown to lie very substantially below the envelope, at least for relatively small values of the number of symmetry-breaking directions under the alternative. In contrast, the asymptotic power of the likelihood ratio test based on the eigenvalues of the sample covariance matrix is shown to be close to that envelope. These results extend to the case of multispiked alternatives the findings of an ear- lier study (Onatski, Moreira and Hallin, 2011) of the single-spiked case. The ∗ Alexei Onatski: University of Cambridge, ao319@cam.ac.uk. Marcelo J. Moreira: Funda¸c˜ ao Getulio Vargas, mjmoreira@fgv.br. Marc Hallin: Universit´e libre de Bruxelles and Princeton University, mhallin@ulb.ac.be. 1 arXiv:1210.5663v1 [math.ST] 20 Oct 2012 Abstract