Stimatori wavelet per la regressione non parametrica in modelli ad effetti misti

The problem of removing noise from a set of known data is classic in the mathematical literature and many methods and papers are devoted to this subject, due to the huge amount of applications that require its solution in different fields (medicine, astronomy, acoustic, and so on). The problem is referred as nonparametric regression to emphasize the objective of finding the best (in some sense) curve that fits data. The aim of this talk is twofold. First we show that a nonparametric linear estimator of a regression function, obtained as solution of some specific regularization problem is also a Bayesian estimator obtained for some particular prior over the regression function and it is the best linear unbiased predictor (BLUP) in a nonparametric mixed effect model. Then, since this estimator is quite heavy to compute, we propose a tight approximation of this estimator, based on the discrete wavelet transform (DWT), easy and fast to implement, which is optimal in the minimax sense for the mean integrated squared error (MISE) for equispaced or non equispaced design.