Non-asymptotic minimax rates of testing in signal detection

Abstract

Let $Y=(Y_i)_{i \in I}$ be a finite or countable sequence of independent Gaussian random variables with mean $f=(f_i)_{i \in I}$ and common variance $\sigma^2$. For various sets $\mathcal{F} \subset \ell_2(I)$, the aim of this paper is to describe the minimal $\ell_2$-distance between $f$ and $0$ for the problem of testing $f=0$ against $f\neq 0$, $f\in \mathcal {F}$, to be possible with prescribed error probabilities. To do so, we start with the set $\mathcal {F}$ which collects the sequences $f$ such that $f_j=0$ for $j>n$ and $ | \{j, f_j \neq 0\}| \leq k$, where the numbers k and n are integers satisfying $1\leq k \leq n$. Then we show how such a result allows us to handle the cases where $\mathcal {F}$ is an ellipsoid and more generally an $\ell_p$-body with $p \in ]0,2]$. Our results are not asymptotic in the sense that we do not assume that $\sigma$ tends to $0$. Finally, we consider the problem of adaptive testing.