2016/02/13 by Cristina Butucea, Butucea, Cristina, Rania Zgheib +1
Mathematics · #62G10 #62H15 #FOS: Mathematics #Point processes and geometric inequalities #Random Matrices and Applications #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #msc:62G10 #msc:62H15 #stat.TH
paper · pdf · doi:10.48550/arxiv.1602.04310
arxiv created 2016/02/13 · openalex publication_date 2016/02/13 · arxiv updated 2016/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 φn,p= (a2n √(p))- (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, ϕn,p= (a2n p)- (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 (a2 n√(p)))α/(4 α+1) and (ln ln (a2 n p))α/(4 α+1) for Toeplitz covariance matrices, respectively.