  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.
