Nonasymptotic One- and Two-Sample Tests in High Dimension with Unknown Covariance Structure

Abstract

Let ${\mathbb {X}}= (X_i)_{1\le i \le n}$ be an i.i.d. sample of square-integrable variables in $\mathbb {R}^d$, with common expectation $\mu$ and covariance matrix $\varSigma$, both unknown. We consider the problem of testing if $\mu$ is $\eta$-close to zero, i.e. $\Vert \mu \Vert \le \eta$ against $\Vert \mu \Vert \ge (\eta + \delta )$; we also tackle the more general two-sample mean closeness (also known as 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 $\big \Vert \mu \big \Vert ^2$ used a test statistic, and secondly for estimating the operator and Frobenius norms of $\varSigma$ 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 $d_* := \big \Vert \varSigma \big \Vert _2^2/\big \Vert \varSigma \big \Vert _\infty ^2$. In particular, for $\eta =0$, the minimum separation distance is ${\varTheta }( d_*^{\nicefrac {1}{4}}\sqrt{\big \Vert \varSigma \big \Vert _\infty /n})$, in contrast with the minimax estimation distance for $\mu$, which is ${\varTheta }(d_e^{\nicefrac {1}{2}}\sqrt{\big \Vert \varSigma \big \Vert _\infty /n})$ (where $d_e:=\big \Vert \varSigma \big \Vert _1/\big \Vert \varSigma \big \Vert _\infty$). This generalizes a phenomenon spelled out in particular by Baraud (Bernoulli 8(5):577–606 2002, [3]).