Estimation of the density of regression errors by pointwise model selection

This paper presents two results: a density estimator and an estimator of regression error density. We first propose a density estimator constructed by model selection, which is adaptive for the quadratic risk at a given point. Then we apply this result to estimate the error density in an homoscedastic regression framework Yi = b(Xi) + ǫi, from which we observe a sample (Xi, Yi). Given an adaptive estimator bb of the regression function, we apply the density estimation procedure to the residuals bǫi = Yi − b(Xi). We get an estimator of the density of ǫi whose rate of convergence for the quadratic pointwise risk is the maximum of two rates: the minimax rate we would get if the errors were directly observed and the minimax rate of convergence of bb for the quadratic integrated risk. February 18, 2009 MSC 2000 Subject Classifications. 62G07-62G08