Nonparametric minimax tests for large covariance matrices - CANCELLED!

Cristina Butucea (Marne-la-Vallée)
Wednesday, June 1, 2016 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We observe n independent p-dimensional Gaussian vectors with missing values, that is each coordinate (which is assumed standardized) is observed with probability a>0. Asymptotically, n and p tend to infinity, a tends to 0. We investigate the test problem of a simple null hypothesis that the high-dimensional covariance matrix of the underlying random vector is the identity matrix (lack of correlations). Under the assumption that the covariance matrix belongs to a Sobolev-type ellipsoid with known 'smoothness' parameter, we give the asymptotic separation rates and see how the parameter a influence the rates. In the particular case a=1, i.e. without missing observations, we describe sharp asymptotic values of the maximal type II error probability. When the 'smoothness' parameter is unknown we propose adaptive test procedure. Our procedures improve on existing minimax methods particularly in the case small n, large p. A different behavior is observed when the covariance matrix is Toeplitz (the case where each vector is issued from a stationary Gaussian process) in the sense that the separation rates are smaller than in the case of general covariance matrices.