  Given an i.i.d. sample $\{(X_i,Y_i)\}_{i \in \{1 \ldots n\}}$ from the random
design regression model $Y = f(X) + \epsilon$ with $(X,Y) \in [0,1] \times
[-M,M]$, in this paper we consider the problem of testing the (simple) null
hypothesis $f = f_0$, against the alternative $f \neq f_0$ for a fixed $f_0 \in
L^2([0,1],G_X)$, where $G_X(\cdot)$ denotes the marginal distribution of the
design variable $X$. The procedure proposed is an adaptation to the regression
setting of a multiple testing technique introduced by Fromont and Laurent
(2005), and it amounts to consider a suitable collection of unbiased estimators
of the $L^2$--distance $d_2(f,f_0) = \int {[f(x) - f_0 (x)]^2 d\,G_X (x)}$,
rejecting the null hypothesis when at least one of them is greater than its
$(1-u_\alpha)$ quantile, with $u_\alpha$ calibrated to obtain a level--$\alpha$
test. To build these estimators, we will use the warped wavelet basis
introduced by Picard and Kerkyacharian (2004). We do not assume that the errors
are normally distributed, and we do not assume that $X$ and $\epsilon$ are
independent but, mainly for technical reasons, we will assume, as in most part
of the current literature in learning theory, that $|f(x) - y|$ is uniformly
bounded (almost everywhere). We show that our test is adaptive over a
particular collection of approximation spaces linked to the classical Besov
spaces.
