Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

TL;DR

This paper establishes non-asymptotic minimax testing rates for goodness-of-fit in heteroscedastic Gaussian models, covering inverse problems and various function spaces without assumptions on variances.

Contribution

It provides the first non-asymptotic minimax rates for testing in heteroscedastic Gaussian sequences, including inverse problems and multiple norms.

Findings

Derived minimax rates for $l_2$ and $l_{}$ norms.

Applicable to finite and countable index sets.

No assumptions on variances or function spaces.

Abstract

The aim of this paper is to establish non-asymptotic minimax rates of testing for goodness-of-fit hypotheses in a heteroscedastic setting. More precisely, we deal with sequences $(Y_j)_{j\in J}$ of independent Gaussian random variables, having mean $(\theta_j)_{j\in J}$ and variance $(\sigma_j)_{j\in J}$. The set $J$ will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to $l_2$ and $l_{\infty}$ norms, without assumption on $(\sigma_j)_{j\in J}$ and on several functions spaces. Our point of view is completely non-asymptotic.