Adaptive test for large covariance matrices in presence of missing observations
Résumé
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 φ~n,p=(a²n–√p)−2α/(4α+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, ϕ~n,p=(a²np)−2α/(4α+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(a²n–√p))α/(4α+1) and (ln ln(a²np))α/(4α+1) for Toeplitz covariance matrices, respectively.