Wavelet Estimators : Adapting to Unknown Smoothness

A wavelet thresholding algorithm is used to recover a function of unknown smoothness from noisy data. It is known that it can be tuned to be minimax in order over a wide range of Besov-type smoothness constraints and L p-losses. We provide a method to estimate an adaptive threshold parameter for each resolution level. It is shown that the proposed algorithm is adaptive in order, i.e. it attains the rate of convergence which is minimax up to a constant over Besov regularity classes and L p-error measures, 1 p 1. The algorithm is computationally straightforward: the whole eeort to compute the threshold is order N log N for the sample size N. Estimateurs par Ondelettes: Adaptation a la R egularit e Inconnue R esum e : On utilise un estimateur par ondelettes dans le cas o u la r egularit e de la fonction a estimer est inconnue. On propose un algorithme de seuillage adaptatif, dans lequel le para-m etre de seuillage est estim e a chaque niveau de r esolution. On d emontre que cet algorithme atteind une vitesse de convergence minimax a une constante pr es pour les risques L p sur les classes de Besov.