An adaptive test for zero mean

Abstract : Assume we observe a random vector y of Rn and write y = f + ε, where f is the expectation of y and ε is an unobservable centered random vector. The aim of this paper is to build a new test for the null hypothesis that f = 0 under as few assumptions as possible on f and ε. The proposed test is nonparametric (no prior assumption on f is needed) and nonasymptotic. It has the prescribed level α under the only assumption that the components of ε are mutually independent, almost surely different from zero and with symmetric distribution. Its power is described in a general setting and also in the regression setting, where fi = F(xi) for an unknown regression function F and fixed design points xi ∈ [0, 1]. The test is shown to be adaptive with respect to H ̈olderian smoothness in the regression setting under mild assumptions on ε. In particular, we prove adaptive properties when the εi's are not assumed Gaussian nor identically distributed.
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Cécile Durot, Yves Rozenholc. An adaptive test for zero mean. Mathematical Methods of Statistics, Allerton Press, Springer (link), 2006, 15 (1), pp.26-60. ⟨hal-00748951⟩

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