On an adaptive goodness-of-fit test with finite sample validity for random design regression models

Given an i.i.d. sample {(Xi,Yi)}i∈{1,...,n} from the random design regression model Y = f(X)+ε with (X,Y) ∈ [0,1]×[−M,M], we consider the problem of testing the (simple) null hypothesis “f = f0”, against the alternative “f ̸= f0” from a non- asymptotic point of view and for a fixed f0 ∈ L2([0,1],GX), where GX(·) denotes the (known) 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 [2], and it amounts to consider a suitable collection of unbiased estimators of the L2–distance d(f,f0), rejecting the null hypothesis when at least one of them is greater than its (1 − uα) quantile, with uα calibrated to obtain a level–α test. These estimators are built upon the warped wavelet system recently introduced by Picard and Kerkyacharian [1], and the resulting testing procedure turns out to be adaptive over a particular collection of approximation spaces linked to the classical Besov spaces. Possible extensions of the proposed procedure to other settings (e.g. two–sample problems, unknown design density, composite-hypotheses), and an analogous goodness–of–fit test on the sphere based on a new type of spherical wavelets, called needlets (see [3]) will also be sketched.