Warped Wavelet and Vertical Thresholding

Let {(Xi, Yi)}i∈{1,...,n} be an i.i.d. sample from the random design regression model Y = f(X) + ε with (X, Y ) ∈ [0, 1] × [−M,M ]. In dealing with such a model, adaptation is naturally to be intended in terms of L([0, 1], GX) norm where GX(·) denotes the (known) marginal distribution of the design variable X. Recently much work has been devoted to the construction of estimators that adapts in this setting (see, for example, [5, 24, 25, 32]), but only a few of them come along with a easy–to–implement computational scheme. Here we propose a family of estimators based on the warped wavelet basis recently introduced by Picard and Kerkyacharian [36] and a tree-like thresholding rule that takes into account the hierarchical (across-scale) structure of the wavelet coefficients. We show that, if the regression function belongs to a certain class of approximation spaces defined in terms of GX(·), then our procedure is adaptive and converge to the true regression function with an optimal rate. The results are stated in terms of excess probabilities as in [19]. AMS 2000 subject classifications: Primary 62G07, 60K35; secondary 62G20.