2008/01/22 by Pierpaolo Brutti, Brutti, Pierpaolo
Mathematics · #60K35 (Primary) #62G07 #62G20 (Secondary) #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #msc:60K35 #msc:62G07 #msc:62G20 #stat.TH
paper · pdf · doi:10.48550/arxiv.0801.3319
Submitted to the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv created 2008/01/22 · arxiv updated 2009/12/01
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 L2([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].