Nonasymptotic one-and two-sample tests in high dimension with unknown covariance structure

Let X = (Xi)1≤ i ≤ n be an i.i.d. sample of square-integrable variables in ℝd, GBwith common expectation μ and covariance matrix Sigma, both unknown. We consider the problem of testing if μ is η-close to zero, i.e. |μ| ≤ η against |μ| ≥ (η + δ); we also tackle the more general two-sample mean closeness (also known as em relevant difference) testing problem. The aim of this paper is to obtain nonasymptotic upper and lower bounds on the minimal separation distance δ such that we can control both the Type I and Type II errors at a given level. The main technical tools are concentration inequalities, first for a suitable estimator of |μ|2 used a test statistic, and secondly for estimating the operator and Frobenius norms of Sigma coming into the quantiles of said test statistic. These properties are obtained for Gaussian and bounded distributions. A particular attention is given to the dependence in the pseudo-dimension d_* of the distribution, defined as d_* := |Sigma|22/|Sigma|_∞2. In particular, for η=0, the minimum separation distance is Theta( d_*^frac14√|Sigma|_∞/n), in contrast with the minimax estimation distance for μ, which is Theta(de^frac12√|Sigma|_∞/n) (where de:=|Sigma|1/|Sigma|_∞). This generalizes a phenomenon spelled out in particular by Baraud (2002).