Fast nonasymptotic testing and support recovery for large sparse Toeplitz covariance matrices

Nayel Bettache, Cristina Butucea and Marianne Sorba

Journal of Multivariate Analysis, 2022, vol. 190, issue C

Abstract: We consider n independent p-dimensional Gaussian vectors with covariance matrix having Toeplitz structure. The aim is two-fold: to test that these vectors have independent components against a stationary distribution with sparse Toeplitz covariance matrix, and also to select the support of non-zero entries under the alternative hypothesis. Our model assumes that the non-zero values occur in the recent past (time-lag less than p/2). We build test procedures that combine a sum and a scan-type procedure, but are computationally fast, and show their non-asymptotic behaviour in both one-sided (only positive correlations) and two-sided alternatives, respectively. We also exhibit a selector of significant lags and bound the Hamming-loss risk of the estimated support. These results can be extended to the case of nearly Toeplitz covariance structure and to sub-Gaussian vectors. Numerical results illustrate the excellent behaviour of both test procedures and support selectors — larger the dimension p, faster are the rates.

Keywords: Covariance matrix; High-dimensional vectors; Hypothesis testing; Sparsity; Support recovery; Time series (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.jmva.2021.104883

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