Signal detection in high dimension
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Date
2012-11-13
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Abstract
This paper deals with the local asymptotic structure, in the sense of
Le Cam’s asymptotic theory of statistical experiments, of the signal detection
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 proposed
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 earlier
study (Onatski, Moreira and Hallin, 2011) of the single-spiked case. The
methods we are using here, however, are entirely new, as the Laplace approximations
considered in the single-spiked context do not extend to the
multispiked case.
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Keywords
asymptotic power, contiguity, large dimensionality, power envelope, sphericity tests, spiked covariance