## Bernoulli

• Bernoulli
• Volume 8, Number 5 (2002), 577-606.

### Non-asymptotic minimax rates of testing in signal detection

Yannick Baraud

#### 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.

#### Article information

Source
Bernoulli, Volume 8, Number 5 (2002), 577-606.

Dates
First available in Project Euclid: 4 March 2004

https://projecteuclid.org/euclid.bj/1078435219

Mathematical Reviews number (MathSciNet)
MR2004e:62014

Zentralblatt MATH identifier
1007.62042

#### Citation

Baraud, Yannick. Non-asymptotic minimax rates of testing in signal detection. Bernoulli 8 (2002), no. 5, 577--606. https://projecteuclid.org/euclid.bj/1078435219

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