Estimation of the density of regression errors by pointwise model selection

Sandra Plancade (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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 $Y_i=b(X_i) + \epsilon _i$, from which we observe a sample $(X_i,Y_i)$. Given an adaptive estimator $\widehat{b}$ of the regression function, we apply the density estimation procedure to the residuals $\widehat{\epsilon} _i = Y_i -\widehat{b} (X_i)$. We get an estimator of the density of $\epsilon _i$ 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 $\widehat{b}$ for the quadratic integrated risk.

Mots clés: density; regression error; pointwise model selection; adaptivity.; adaptivity

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Sandra Plancade. Estimation of the density of regression errors by pointwise model selection. 2009. ⟨hal-00364334⟩