Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

JOURNAL OF MULTIVARIATE ANALYSIS

卷:146

Article

Zgheib, Rania¹ 

[1] Univ Paris Est Marne la Vallee, LAMA, UPEMLV, UMR 8050, F-77454 Marne La Vallee, France

关键词: Toeplitz matrix;    Covariance matrix;    High-dimensional data;    U-statistic;    Minimax hypothesis testing;    Optimal separation rates;    Sharp asymptotic rates;   

DOI  :  10.1016/j.jmva.2015.09.003

来源: Elsevier

【 摘 要 】

We observe a sample of n independent p-dimensional Gaussian vectors with Toeplitz covariance matrix Sigma = [sigma(vertical bar i-j vertical bar)](1 <= i,j <= p) and sigma(0) = 1. We consider the problem of testing the hypothesis that Sigma is the identity matrix asymptotically when n -> infinity and p -> infinity. We suppose that the covariances sigma(k) decrease either polynomially (Sigma(k >= 1) k(2 alpha)sigma(2)(k) <= L for alpha > 1/4 and L > 0) or exponentially (Sigma(k >= 1) e(2Ak)sigma(2)(k) <= L for A, L > 0). We consider a test procedure based on a weighted U-statistic of order 2, with optimal weights chosen as solution of an extremal problem. We give the asymptotic normality of the test statistic under the null hypothesis for fixed n and p -> +infinity and the asymptotic behavior of the type I error probability of our test procedure. We also show that the maximal type II error probability, either tend to 0, or is bounded from above. In the latter case, the upper bound is given using the asymptotic normality of our test statistic under alternatives close to the separation boundary. Our assumptions imply mild conditions: n = o(p(2 alpha-1/2)) (in the polynomial case), n = o(e(p)) (in the exponential case). We prove both rate optimality and sharp optimality of our results, for alpha > 1 in the polynomial case and for any A > 0 in the exponential case. A simulation study illustrates the good behavior of our procedure, in particular for small n, large p. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free