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Given an i.i.d. sample $\{(X_i,Y_i)\}_{i \in \{1 \ldots n\}}$ from the randomdesign 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) nullhypothesis $f = f_0$, against the alternative $f \neq f_0$ for a fixed $f_0 \inL^2([0,1],G_X)$, where $G_X(\cdot)$ denotes the marginal distribution of thedesign variable $X$. The procedure proposed is an adaptation to the regressionsetting of a multiple testing technique introduced by Fromont and Laurent(2005), and it amounts to consider a suitable collection of unbiased estimatorsof 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 basisintroduced by Picard and Kerkyacharian (2004). We do not assume that the errorsare normally distributed, and we do not assume that $X$ and $\epsilon$ areindependent but, mainly for technical reasons, we will assume, as in most partof the current literature in learning theory, that $|f(x) - y|$ is uniformlybounded (almost everywhere). We show that our test is adaptive over aparticular collection of approximation spaces linked to the classical Besovspaces.

A note on an Adaptive Goodness-of-Fit test with Finite Sample Validity for Random Design Regression Models