Adaptive test for large covariance matrices with missing observations

TL;DR

This paper develops adaptive testing procedures for high-dimensional covariance matrices with missing data, establishing minimax rates and showing how missingness affects test performance.

Contribution

It introduces asymptotically minimax tests for covariance matrices with missing observations, including adaptive methods that do not require prior knowledge of smoothness parameter.

Findings

Derived minimax separation rates for general and Toeplitz covariance matrices.

Proposed adaptive tests with near-optimal rates, accounting for missing data.

Quantified the impact of missingness parameter on testing rates.

Abstract

We observe $n$ independent $p-$dimensional Gaussian vectors with missing coordinates, that is each value (which is assumed standardized) is observed with probability $a>0$. We investigate the problem of minimax nonparametric testing that the high-dimensional covariance matrix $\Sigma$ of the underlying Gaussian distribution is the identity matrix, using these partially observed vectors. Here, $n$ and $p$ tend to infinity and $a>0$ tends to 0, asymptotically. We assume that $\Sigma$ belongs to a Sobolev-type ellipsoid with parameter $\alpha >0$. When $\alpha$ is known, we give asymptotically minimax consistent test procedure and find the minimax separation rates $\tilde \varphi_{n,p}= (a^2n \sqrt{p})^{- \frac{2 \alpha}{4 \alpha +1}}$, under some additional constraints on $n,\, p$ and $a$. We show that, in the particular case of Toeplitz covariance matrices,the minimax separation rates are faster, $\tilde \phi_{n,p}= (a^2n p)^{- \frac{2 \alpha}{4 \alpha +1}}$. We note how the "missingness" parameter $a$ deteriorates the rates with respect to the case of fully observed vectors ($a=1$). We also propose adaptive test procedures, that is free of the parameter $\alpha$ in some interval, and show that the loss of rate is $(\ln \ln (a^2 n\sqrt{p}))^{\alpha/(4 \alpha +1)}$ and $(\ln \ln (a^2 n p))^{\alpha/(4 \alpha +1)}$ for Toeplitz covariance matrices, respectively.