文章基本信息

标题：Adaptive test for large covariance matrices in presence of missing observations
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作者：Cristina Butucea ; Rania Zgheib
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2017
卷号：XIV
页码：557-578
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We observe n independent p−dimensional Gaussian vectors with missingcoordinates, that is each value (which is assumed standardized) is observedwith probability a > 0. We investigate the problem of minimax nonparametrictesting that the high-dimensional covariance matrix of the underlying Gaussiandistribution is the identity matrix, using these partially observed vectors. Here, nand p tend to infinity and a > 0 tends to 0, asymptotically.We assume that belongs to a Sobolev-type ellipsoid with parameter α > 0.When α is known, we give asymptotically minimax consistent test procedure andfind the minimax separation rates ˜ϕn,p = (a2n√p)− 24+1 , under some additionalconstraints on n, p and a. We show that, in the particular case of Toeplitz covariancematrices, the minimax separation rates are faster, ˜φn,p = (a2np)− 24+1 . Wenote how the ”missingness” parameter a deteriorates the rates with respect to thecase of fully observed vectors (a = 1).We also propose adaptive test procedures, that is free of the parameter α insome interval, and show that the loss of rate is (ln ln(a2n√p))α/(4α+1) in general,and (ln ln(a2np))α/(4α+1) for Toeplitz covariance matrices, respectively.
关键词：adaptive test; covariance matrices; goodness-of-fit tests; minimax;separation rate; missing observation; Toeplitz matrices.