Testing in High-Dimensional Spiked Models
Authors
Johnstone, I. M
Onatski, A.
Publication Date
2018-01-25Series
Cambridge Working Papers in Economics
Type
Working Paper
Metadata
Show full item recordCitation
Johnstone, I. M., & Onatski, A. (2018). Testing in High-Dimensional Spiked Models. https://doi.org/10.17863/CAM.21781
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 {code} 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.
Keywords
Likelihood ratio test, hypergeometric function, principal components analysis, canonical correlations, matrix denoising, multiple response regression
Identifiers
CWPE1806
This record's DOI: https://doi.org/10.17863/CAM.21781
This record's URL: https://www.repository.cam.ac.uk/handle/1810/274648
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