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

Nayel Bettache, Cristina Butucea, Marianne Sorba

Abstract: We consider n<math><mi is="true">n</mi></math> independent p<math><mi is="true">p</mi></math>-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<math><mrow is="true"><mi is="true">p</mi><mo is="true">/</mo><mn is="true">2</mn></mrow></math>). 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<math><mi is="true">p</mi></math>, faster are the rates.

External IDs:dblp:journals/ma/BettacheBS22