Sharp minimax tests for large Toeplitz covariance matrices with repeated observations

TL;DR

This paper develops sharp minimax tests for high-dimensional Toeplitz covariance matrices, demonstrating their optimality and effectiveness in detecting deviations from the identity matrix under polynomial and exponential decay conditions.

Contribution

It introduces a new weighted U-statistic based testing procedure with proven asymptotic normality and optimality for large Toeplitz covariance matrices with specific decay rates.

Findings

Test statistic is asymptotically normal under null hypothesis.

Maximal type II error tends to zero or is bounded, depending on the alternative.

Procedure performs well even with small sample size and large dimension.

Abstract

We observe a sample of $n$ independent $p$-dimensional Gaussian vectors with Toeplitz covariance matrix $ \Sigma = [\sigma_{|i-j|}]_{1 \leq i,j \leq p}$ and $\sigma_0=1$. We consider the problem of testing the hypothesis that $\Sigma$ is the identity matrix asymptotically when $n \to \infty$ and $p \to \infty$. We suppose that the covariances $\sigma_k$ decrease either polynomially ($\sum_{k \geq 1} k^{2\alpha} \sigma^2_{k} \leq L$ for $ \alpha >1/4$ and $L>0$) or exponentially ($\sum_{k \geq 1} e^{2Ak} \sigma^2_{k} \leq L$ 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 $p \to + \infty$ 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 $p$.