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Adaptive test for large covariance matrices with missing observations

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 $Σ$ 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 $Σ$ belongs to a Sobolev-type ellipsoid with parameter $α>0$. When $α$ is known, we give asymptotically minimax consistent test procedure and find the minimax separation rates $\tilde φ_{n,p}= (a^2n \sqrt{p})^{- \frac{2 α}{4 α+1}}$, 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, $\tilde ϕ_{n,p}= (a^2n p)^{- \frac{2 α}{4 α+1}}$. 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 $α$ in some interval, and show that the loss of rate is $(\ln \ln (a^2 n\sqrt{p}))^{α/(4 α+1)}$ and $(\ln \ln (a^2 n p))^{α/(4 α+1)}$ for Toeplitz covariance matrices, respectively.