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

Abstract

Let X = (Xi) 1≤i≤n be an i.i.d. sample of square-integrable variables in R d , with common expectation µ and covariance matrix Σ, 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 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 Σ 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 asIn particular, for η = 0, the minimum separation distance is Θ(d 1/4 * Σ ∞ /n), in contrast with the minimax estimation distance for µ, which is Θ(d 1/2 e Σ ∞ /n) (where de := Σ 1 / Σ ∞ ). This generalizes a phenomenon spelled out in particular by Baraud (2002).

An Rkhs Approach for Pivotal Inference in Functional Linear Regression

“…Moreover, in the context of drug development, several authors considered relevant hypotheses for comparing dose response files (see Liu et al, 2007Liu et al, , 2009, where they estimate parametric curves from real valued data. On the other hand, hypotheses of the form (1.3) have found considerable interest in mathematical statistics, see Spokoiny (1996) or Lepski and Spokoiny (1999) for some early and Blanchard and Fermanian (2021); Brutsche and Rohde (2022) for some more recent references.…”