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Optimal hypothesis testing for high dimensional covariance matrices. (English) Zbl 1281.62140

Summary: This paper considers testing a covariance matrix \(\Sigma\) in a high dimensional setting where the dimension \(p\) can be comparable or much larger than the sample size \(n\). The problem of testing the hypothesis \(H_{0}:\Sigma=\Sigma_{0}\) for a given covariance matrix \(\Sigma_{0}\) is studied from a minimax point of view. We first characterize the boundary that separates the testable regions from the non-testable regions 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 in 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.

MSC:

62H15 Hypothesis testing in multivariate analysis
62C20 Minimax procedures in statistical decision theory
62F05 Asymptotic properties of parametric tests
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