Testing in high-dimensional spiked models

By Iain M. Johnstone, A. Onatski. Annals of Statistics, 2015

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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.

Subfields: Mathematical Physics, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/78ed3332b2ee07b9681bfce32e8801084dcd4c7b

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Citations in the corpus

Fundamental limits of detection in the spiked Wigner model (2018, background): “Important progress on the detection problem was made by Onatski et al. (2013, 2014) and Johnstone and Onatski (2015), who considered the spiked covariance model for a uniformly distributed unit norm spike, and studied the asymptotics of the likelihood ratio (LR) of a spiked alternative against a…”