FULLY DATA-DRIVEN NONPARAMETRIC VARIANCE ESTIMATORS

We consider the problem of estimating the unknown variance function up silon in a nonparametric regression model. As a basis for our estimato rs we take estimated residuals which are based on a kernel estimator o f the mean vector. Then we form with these residuals a kernel estimato r of upsilon. Main emphasis is on a data-driven choice of the bandwidt hs involved in the procedure. It is shown that the risk of this estima tor attains the uniform convergence rate in Sobolev classes for upsilo n under weak smoothness assumptions on the mean. Moreover, we prove th at there is asymptotically no loss due to the estimation of the mean. AMS 1991 Mathematics subject classification: Primary 62G07; secondary 62G20.