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

Let $\mathbf{X} = (X_i)_{1\leq i \leq n}$ be an i.i.d. sample of square-integrable variables in $\mathbb{R}^d$, with common expectation $\mu$ and covariance matrix $\Sigma$, both unknown. We consider the problem of testing if $\mu$ is $\eta$-close to zero, i.e. $\|\mu\| \leq \eta$ against $\|\mu\| \geq (\eta + \delta)$; we also tackle the more general two-sample mean closeness testing problem. The aim of this paper is to obtain nonasymptotic upper and lower bounds on the minimal separation distance $\delta$ 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 $\|\mu\|^2$ 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 $d_* := \|\Sigma\|_2^2/\|\Sigma\|_\infty^2$. In particular, for $\eta=0$, the minimum separation distance is ${\Theta}(d_*^{\frac{1}{4}}\sqrt{\|\Sigma\|_\infty/n})$, in contrast with the minimax estimation distance for $\mu$, which is ${\Theta}(d_e^{\frac{1}{2}}\sqrt{\|\Sigma\|_\infty/n})$ (where $d_e:=\|\Sigma\|_1/\|\Sigma\|_\infty$). This generalizes a phenomenon spelled out in particular by Baraud (2002).