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

Abstract

The aim of this paper is to establish non-asymptotic minimax rates for goodness-of-fit hypotheses testing in an heteroscedastic setting. More precisely, we deal with sequences (Y j ) j∈J of independent Gaussian random variables, having mean (θ j ) j∈J and variance (σ j ) j∈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 ∞ norms, without assumption on (σ j ) j∈J and on several functions spaces. Our point of view is entirely non-asymptotic.

Testing inverse problems: A direct or an indirect problem?

“…The corresponding rates of testing have been computed very recently in Ingster et al (2010) or Laurent et al (2010) and different testing procedures have been proposed. We may alternatively consider the hypothesis image of the assumption H 0 IP by the operator T…”

“…For more details, we refer for instance to Baraud (2002) in the direct case (i.e. when T denotes the identity) or to Laurent et al (2010) in an heteroscedastic setting, which is equivalent to the inverse problem modeled in (3).…”

“…They are given in Laurent et al (2010) and recalled in Table 1. In the following, we write m k $ n k if there exists two constants c 1 and c 2 such that for all k Z1, c 1 rm k =n k rc 2 .…”

“…As said before in Section 2, choosing the best framework for building signal detection tests involves knowing the minimax rates of testings for the both settings: the direct and the indirect version of the observation model. On the one hand, rates of testing for the inverse problem (3) have been obtained in Laurent et al (2010) under such assumptions. For C L ða,bÞ,C U ða,bÞ two positive constants, we have…”

“…If some particular cases such as convolution problems have been widely investigated (see Butucea et al, 2009or Holzmann et al, 2007 for general references), the general case of tests for inverse problem have only been tackled very recently. We refer to Ingster et al (2010) and Laurent et al (2010) for a complete asymptotical and non-asymptotical theory. The previous dilemma concerning the choice of the space where to perform the study is here also crucial.…”