Asymptotic power of sphericity tests for high-dimensional data
By A. Onatski, Marcelo J. Moreira, Marc Hallin. 2013
153 citations recorded by the source corpus.
This paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity. We establish the convergence, under the null hypothesis and the alternative, of the log ratio of the joint densities of the sample covariance eigenvalues to a Gaussian process indexed by the norm of the perturbation. When the perturbation norm is larger than the phase transition threshold studied in Baik et al. (2005), the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the process is non-degenerate, so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous. Using the asymptotic theory of statistical experiments, we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region. In particular, we show that the asymptotic power of the Tracy-Widom-type tests is trivial, whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.
View Paper: https://www.semanticscholar.org/paper/5eb1977779fc382ed1a8bb49b9d7a794037f66ba
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Citations in the corpus
- On the sphericity test with large-dimensional observations (2013, background): “5) reduces to a result provided in [20].”
- A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, methodology): “For example, the sphericity test based on arithmetic mean of the eigenvalues and maximum likelihood ratio principle needs either Gaussian assumption or high moments assumption [24, 41, 55].”
- Optimal hypothesis testing for high dimensional covariance matrices (2012, methodology): “In this asymptotic regime, Theorem 7 in [15] gives a lower bound result analogous to Theorem 1.”
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “Not trying to be comprehensive, one can refer to [8, 12, 28, 40, 44] for instance.”
- Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, background): “But unlike estimation problems there are only few optimality results on testing, see Baik et al. (2005), El Karoui (2007), Cai and Ma (2013) and Onatski et al. (2013).”
- Testing in high-dimensional spiked models (2015, methodology): “The sequences of measures corresponding to the distributions of Λ under the null and under super-critical alternatives are asymptotically mutually singular as n, p→γ ∞ (see Montanari et al (2014) and Onatski et al (2013) for a detailed analysis of SMD and PCA cases).”
- 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…”
- The spectral norm of random inner-product kernel matrices (2015, background): “However, in the “subcritical” regime λ < λ ∗, |v̂T v| → 0 almost surely, λmax(Σ̂) cannot distinguish the null and spiked models, and furthermore no test using only the eigenvalues of Σ̂ can distinguish the models with probability approaching one [32, 3, 4, 44, 42, 43].”