Estimation of the density of regression errors by pointwise model selection
Résumé
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 a homoscedastic regression framework Y i = b(X i ) + ε i from which we observe a sample (X i , Y i ). Given an adaptive estimator b^ of the regression function, we apply the density estimation procedure to the residuals ε^i=Yi−b^(Xi) . We get an estimator of the density of ε i 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 b^ for the quadratic integrated risk.