Adaptive test for large covariance matrices in presence of 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 (Formula presented), under some additional constraints on n, p and a. We show that, in the particular case of Toeplitz covari-ance matrices, the minimax separation rates are faster, (Formula presented). 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 (Formula presented) in general, and (lnln(a2np))α/(4α+1) for Toeplitz covariance matrices, respectively.