Asymptotic Power of Sphericity Tests for High-dimensional Data and Multispiked Alternatives

Abstract

This paper extends Onatski, Moreira and Hallin's (2011) study of the power of high-dimensional sphericity tests to the case of multiple symmetry- breaking directions. Simple analytical expressions for the asymptotic power envelope and the asymptotic powers of previously proposed tests are de- rived. 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 data reduced to the eigenvalues of the sample covariance matrix is shown to be close to that envelope. Key words: sphericity tests, large dimensionality, asymptotic power, spiked covariance, contiguity, power envelope.