Testing in high-dimensional spiked models
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Publication Date
2020-06-01Journal Title
Annals of Statistics
ISSN
0090-5364
Publisher
Institute of Mathematical Statistics
Volume
48
Issue
3
Pages
1231-1254
Type
Article
This Version
AM
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Johnstone, I., & Onatskiy, A. (2020). Testing in high-dimensional spiked models. Annals of Statistics, 48 (3), 1231-1254. https://doi.org/10.1214/18-AOS1697
Abstract
We consider the five classes of multivariate statistical problems identified
by James (1964), which together cover much of classical multivariate analysis,
plus a simpler limiting case, symmetric matrix denoising. Each of James'
problems involves the eigenvalues of $E^{-1}H$ where $H$ and $E$ are
proportional to high dimensional Wishart matrices. Under the null hypothesis,
both Wisharts are central with identity covariance. Under the alternative, the
non-centrality or the covariance parameter of $H$ has a single eigenvalue, a
spike, that stands alone. When the spike is smaller than a case-specific phase
transition threshold, none of the sample eigenvalues separate from the bulk,
making the testing problem challenging. Using a unified strategy for the six
cases, we show that the log likelihood ratio processes parameterized by the
value of the sub-critical spike converge to Gaussian processes with logarithmic
correlation. We then derive asymptotic power envelopes for tests for the
presence of a spike.
Identifiers
External DOI: https://doi.org/10.1214/18-AOS1697
This record's URL: https://www.repository.cam.ac.uk/handle/1810/287808
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