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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 (Yj)j∈J of independent Gaussian random variables, having mean (θj)j∈J and variance (σj)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 l2 norm, without assumption on (σj)j∈J and on several functions spaces. Our point of view is entirely non-asymptotic.
Highlights
We consider the following heteroscedastic statistical model : Yj = θj + σjǫj, j ∈ J, (1.1)where θ =j∈J is unknown, σ =j∈J is assumed to be known, and the variablesj∈J are i.i.d. standard normal variables
The set J is either {1, . . . , N } for some N ∈ N∗ or N∗
The particular case σj = σ for all j ∈ J corresponds to the classical statistical model where the variance of the observations is always the same
Summary
The particular case σj = σ for all j ∈ J corresponds to the classical statistical model where the variance of the observations is always the same It has been widely considered in the literature, both for test and estimation approaches. The aim of the paper is to determine this minimax rate of testing over various classes of alternatives F, for the test of null hypothesis ”θ = 0” in Model (1.1) with respect to the l2 and l∞ norms. The main reference for computing minimax rates of testing over non parametric alternatives is the series of paper due to Ingster[11], where various statistical models and a wide range of sets of alternatives are considered.
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