An Adaptive Test for Zero Mean

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 Hölderian 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.