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

Abstract: Let $\mathbf{X} = (X_i)<em>{1\leq i \leq n}$ be an i.i.d. sample of square-integrable variables in $\mathbb{R}<sup>d$, \GB{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 (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 $\delta$ 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 $|\mu|<sup>2$ 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 $d</em><em>$ of the distribution, defined as $$d_</em> := |\Sigma|<em>2^2/|\Sigma|</em>\infty<sup>2$$. In particular, for $\eta=0$, the minimum separation distance is $${\Theta}( d_*<sup>{\frac{1}{4}}\sqrt{|\Sigma|_\infty/n})$$, in contrast with the minimax estimation distance for $\mu$, which is $${\Theta}(d_e<sup>{\frac{1}{2}}\sqrt{|\Sigma|_\infty/n})$$ (where $d_e:=|\Sigma|<em>1/|\Sigma|</em>\infty$). This generalizes a phenomenon spelled out in particular by Baraud (2002).