Optimal hypothesis testing for high dimensional covariance matrices

By Tommaso Cai, Zongming Ma. 2012

117 citations recorded by the source corpus.

This paper considers testing a covariance matrixin the high dimensional setting where the dimension p can be comparable or much larger than the sample size n. The problem of testing the hypothesis H0 : � = � 0 for a given covariance matrix � 0 is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius norm when the ratio between the dimension p over the sample size n is bounded. A test based on a U -statistic is introduced and is shown to be rate optimal over this asymptotic regime. Furthermore, it is shown that the power of this test uniformly dominates that of the corrected likelihood ratio test (CLRT) over the entire asymptotic regime under which the CLRT is applicable. The power of the U -statistic based test is also analyzed when p/n is unbounded.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/43c69135016d6db3621ae722c77aa695f0f23542

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Statistics and Probability

Citations in the corpus

Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, background): “In a separate paper, Cai and Ma (2013) showed that this procedure is indeed optimal under some natural alternatives.”
Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing (2014, methodology): “In another perspective, a minimax analysis for the one-sample identity test x = Ip has been recently proposed in [11].”
Testing independence in high dimensions with sums of rank correlations (2015, methodology): “Despite this mismatch, the comparable power of the test of Cai and Ma (2013) indicates that the three statistics Sτ , Tτ and Sr are all powerful in detecting the signal ‖R − Im‖2 ‖ τ‖2; recall that our experiment has with 1’s on the diagonal so that = R. Lastly, we speculate that Sr is minimax…”
Testing mutual independence in high dimension via distance covariance (2016, methodology): “In this subsection, we study the boundary for the testable and non-testable region and conduct power analysis for our test from a minimax point of view following the work of Cai and Ma (2013), and we also show that our test is rate optimal. We focus on the case where W = .W.1/, : : : , W/ follows a ”
Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, background): “Cai and Ma (2013) characterized the optimal testing boundary that separates the testable region from the non-testable region by the Frobenius norm in the setting where p/n is bounded.”