2015/06/04 by Cristina Butucea, Butucea, Cristina, Rania Zgheib +1
Mathematics · #FOS: Mathematics #Mathematical Inequalities and Applications #Point processes and geometric inequalities #Random Matrices and Applications #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1506.01557
arxiv created 2015/06/04 · openalex publication_date 2015/06/04 · arxiv updated 2015/06/05 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We observe a sample of n independent p-dimensional Gaussian vectors with Toeplitz covariance matrix Σ= [σ|i-j|]1 ≤ i,j ≤ p and σ0=1. We consider the problem of testing the hypothesis that Σ is the identity matrix asymptotically when n → ∞ and p → ∞. We suppose that the covariances σk decrease either polynomially (∑k ≥ 1 k2α σ2k ≤ L for α>1/4 and L>0) or exponentially (∑k ≥ 1 e2Ak σ2k ≤ 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 → + ∞ 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(p2α- 1/2) (in the polynomial case), n=o(ep) (in the exponential case). We prove both rate optimality and sharp optimality of our results, for α>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.